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C. A. Taylor 



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THE WYKEHAM SCIENCE SERIES 



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The Wykeham Science Series 



General Editors: 

PROFESSOR SIR NEVILL MOTT, F.R.S. 

Emeritus Cavendish Professor of Physics 
University of Cambridge 

G. R. NOAKES 

Formerly Senior Physics Master 
Uppingham School 



The Author: 

Charles a. taylor worked in the Admiralty during the War, and 
then joined the University of Manchester Institute of Science and 
Technology. Since 1965 he has been Professor of Physics at University 
College, Cardiff and is also Professor of Experimental Physics at 
the Royal Institution of Great Britain. He has devoted much of his 
research career to the development of visual analogue approaches to 
the interpretation of X-ray and electron diffraction patterns, and in 
addition takes a lively interest in physics education in general. 

The Schoolmaster: 

G. e. foxcroft, educated at Trinity College, Cambridge, is Senior 
Science Master at Rugby School. He has been associated with 
the Nuffield O level Physics Project and was a team member 
for the Nuffield A level Physics Project. 



IMAGES 



A unified view of diffraction 
and image formation with al 
kinds of radiation 



CHARLES A. TAYLOR 

University College, Cardiff 



ACCESSION No. . ~ c - . 


class no. 

S3*, 3a -ms/ 


2 AUG 1979 


o.o 


N 

y 


CAIEGORY 




iJE 



WYKEHAM PUBLICATIONS (LONDON) LTD 
(A member of the Taylor & Francis Group) 
LONDON and BASINGSTOKE 1978 



First published 1978 by Wykeham Publications (London) Ltd. 

© 1978 C. A.Taylor 

All rights reserved. No part of this publication may be reproduced, 
stored in a retrieval system, or transmitted, in any form or by any 
means, electronic, mechanical, photocopying, recording or otherwise, 
without the prior permission of the copyright owner. 



ISBN 85109 620 4 (paper) 
ISBN 85109 630 1 (cloth) 



Printed in Great Britain by Taylor & Francis (Printers) Ltd., 
Rankine Road, Basingstoke, Hants RG24 OPR. 



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Contents 



Preface vii 

1. The physics involved in forming an image 

1.1 What are the common features of all image-forming 
systems ? 1 

1.2 What are the limitations which restrict the precision 

of an image? 7 

1.3 To what kinds of radiation do these principles apply? 10 

2. A closer look at radiation, scattering and diffraction 

2.1 Radiation and the idea of coherence 13 

2.2 Coherence in practice 18 

2.3 Coherence, bandwidth and the Uncertainty Principle 21 

2.4 The nature of the scattering process 24 

2.5 Nomenclature of diffraction and scattering processes 35 

2.6 How can we observe optical diffraction patterns? 40 

2.7 How can we record scattering patterns for non-visible 
radiations? 47 

/ 2.8 How can we measure coherence? 51 

2.9 Self-luminous and incoherently illuminated objects 54 

3. Principles of direct recombination processes 

3.1 ' Straight line ' imaging with pinhole cameras 55 

3.2 Field emission microscopes 61 

3.3 Lenses for light 65 

3.4 A hybrid technique — the zone plate 71 

3.5 Lenses for electrons 76 

3.6 Imaging by scanning 77 






CONTENTS 

4. Principles of indirect recombination processes 

4. 1 The essential problem in X-ray, neutron and electron 
diffraction 81 

4.2 Possible solutions using analogue methods 89 

4.3 Holography 102 



5. Perturbations of the image 

5.1 Aperture and wavelength 

5.2 False detail and possible misrepresentation near the 



108 





resolution limit 


115 


5.3 Abbe's theory of microscopic vision 


120 


5.4 


Image ' formants ' — the fingerprint of the apparatus 


122 


5.5 


Methods of measuring the performance of imaging 






systems 


124 


5.6 


Depth of focus problems 


124 


5.7 Conclusion 


128 


6. Applications and results 




6.1 


Introduction 


129 


6.2 


The eye 


129 


6.3 


The camera 


131 


6.4 


Optical telescopes 


140 


6.5 


Radio telescopes 


149 


6.6 


Schlieren techniques 


151 


6.7 


Image processing 


154 


6.8 


Optical microscopy 


157 


6.9 


Electron microscopy 


162 


6.10 X-ray, electron and neutron diffraction 


170 


6.11 


Radar 


180 


6.12 Medical imaging techniques, including the use of 






X-rays, ultrasonics, y-rays and infrared rays 


186 


6.13 


Holography 


198 


Further reading 


201 


Index 




202 



VI 



Preface 



Two main scientific themes have been at the centre of my research and, 
by an interesting — but not altogether surprising — coincidence there 
are strong mathematical links between them. The topics are the 
diffraction of electromagnetic waves and the physics of musical 
instruments, and the mathematical concept that connects them is that 
of Fourier transformation. I am not, however, the kind of physicist 
for whom the mathematical equations immediately evoke the under- 
lying physics. I accept that mathematics is vital in developing and 
establishing a theory — but 1 hold the view that, once established, it 
ought to be possible to present it in a meaningful way without high- 
level mathematics. This view, together with the notion that science 
is a good deal more unified than our traditional fragmented teaching 
methods sometimes imply, are two philosophical ideas that seem to 
pervade my thinking and I feel sure that they have coloured my 
my approach to this book. 

My theme is the unity of all diffraction and imaging processes, and I 
have tried to present what is, in effect, a study of certain practical 
aspects of the Fourier transform concept as it is applied in modern 
optics, with practically no mathematics. Here and there I have 
included a section, marked with an asterisk, that gives a little back- 
ground mathematics but this is merely for the benefit of those who 
seek it; to omit these sections should not disturb the continuity. 

The book is intended — as are all books in this series — primarily for 
sixth-form students. I hope, however, that it may find a place as a 
background reader in physics courses at University and Polytechnic 
level. I hope too that students of many other sciences at different 
levels who need to use image-forming systems or diffraction techniques 
without necessarily going through the discipline of a physics course 
will find something of interest here. In particular, it may help in 
understanding the potential and the limitations of the various techniques 
described. 



vn 



PREFACE 

In writing the book, I have drawn freely on my memory of books 
read, lectures heard, and demonstrations seen; it is all too easy to 
remember the substance and to forget the source. I am grateful to 
many people mentioned specifically but I am also greatly indebted — 
and apologetic — to those whose contributions are not identified. 

Professor Henry Lipson undoubtedly provided the inspiration for my 
approach to the subject and I owe a great deal to his interest and 
guidance. 

I am grateful to Mr. Noakes, general editor of this series, for planting 
the idea of writing on this theme. I was fortunate in being invited 
to lecture to schools at the Royal Institution on this topic and the 
search for demonstration material for that lecture yielded the basic 
collection round which the illustrations to this book have been 
produced. My thanks therefore go to Sir George Porter and Professor 
Ronald King for providing that opportunity. 

Mr. Geoffrey Foxcroft has been my collaborating schoolmaster and 
has not only fulfilled the scheduled function of making sure that the 
academic member of the partnership did not take off into the clouds 
but has also commented, criticized and contradicted in a most rewarding 
and helpful way. It is a great comfort to know that he has read the 
manuscript with a most school-masterly eye for detail as well as for 
the broad issues. Nevertheless, any errors and heresies that un- 
doubtedly will be discovered must remain my sole responsibility. 

Bob Watkins, the photographer in my department, has played a very 
significant part in producing a high proportion of the illustrations and 
my colleagues George Harburn, Alan Fowler and Mike Evans have 
also given valuable help in preparing illustrations. My thanks go to 
all these people as well as to those who have agreed to let me use their 
illustrations (acknowledgments of which are given elsewhere), to 
Mr. W. Sutherland of Velindre Hospital and Professor Evans of the 
Welsh National School of Medicine for providing illustrations — some 
of which were specially produced, to Jayne Tonks, an electronics 
technician in the department who allowed herself to be photographed 
and to various colleagues in my department who have helped me to 
clarify various ideas in discussion; Dr. Barrie Jones and his colleagues 
at the Open University have drawn me into helpful discussions of a 
course that they have prepared on a closely related theme; my secretary 
Mrs. Valerie Chown deciphered my manuscript in her usual helpful 
way; and finally my gratitude goes to my wife and family for their 
patient forbearance and encouragement. 

CHARLES TAYLOR 

viii 



1. The physics involved in 
forming an image 



•' However entrancing it is to wander unchecked through a 
garden of bright images, are we not enticing your mind from 
another subject of almost equal importance? " 

Ernest Bramah 

Kai Lung's Golden Hours 



1 . 1 . What are the common features of all image-forming systems ? 

We all possess image-forming devices as part of our anatomy, and a 
study of the precise way in which our eyes operate will give us a good 
basis for starting our discussions. Many textbooks — at all levels — 
have been written which include detailed accounts of the physiological 
structure of the eye, or of the paths traced by the rays of light through 
the lens and the various media with which the eye is filled. I do not 
propose to follow this pattern at all; we shall not consider detailed 
operation but rather the broad essential functions that must be 
performed in order to see. 

Imagine a lecturer standing in front of his audience. He is clearly 
gaining information continuously through his eyes. One student has a 
rather florid tie; a woman is examining the state of her make-up in a 
mirror; a man is having trouble with his fountain pen; two people in 
the back row have dozed off to sleep, and so on. Similarly the 
audience receives information about the lecturer. This information 
is, of course, conveyed from one to the other via the light in the room. 
That is a very obvious statement since, if we put the room in total 
darkness, the transfer of information ceases immediately. (In fact to 
put the room in total darkness is suggested as the first experiment 
which dramatically draws attention to the first part of the image- 
forming process.) What happens normally is that the light in the 
room falls on the various objects in it and is both scattered and 
modified; as we shall see later, the scattering depends on the shape 
and texture, the colour may be modified by selective absorption of 



2 IMAGES 

different wavelengths, and there may also be changes in polarization. 
It is the scattering process, however, that will be of most concern to 
us in this book. 

The word ' scatter ' needs considerable elaboration and indeed, a 
large part of Chapter 2 is devoted to it. For the moment it will suffice 
if we interpret it as meaning that the waves bounce off the object in 
some way and hence the original pattern of waves is disturbed by the 
presence of the object. 

The next point to recognize is that only a tiny fraction of this 
scattered light ever reaches the eye of the observer but, nevertheless, 
his eye and brain together are able to abstract an extraordinary amount 
of information from it. That all the information is present in the 
scattered light may easily be demonstrated by a second ' silly ' experi- 
ment: a small card is placed in front of the eye. Obviously the infor- 
mation transfer ceases since the waves are prevented from entering the 
eye; the waves that carried the information must therefore be falling 
on the card. But, if the card were replaced by a photographic plate, 
the resulting record would clearly not be useful; it would just be more 
or less uniform blackness. Nevertheless, all the information must be 
potentially available at that point and later we shall need to consider 
how it could be extracted and in what form it exists. 

A third, and probably the most useful, of our introductory ' silly ' 
experiments which drives this point home is to place a rather improbable 
slide (I often use a cartoon or a very ancient poster — in fact anything 
that the audience is unlikely to expect) in a projector without the lens 
in position. The patch produced on the screen is then completely 
uninformative and yet a moment's thought makes it clear that all the 
necessary information for the production of a complete image of the 
slide must be present on the screen, though in a form which is not 
directly interpretable. Fortunately in this particular example all that 
is necessary to interpret the pattern is to replace the lens in the projector. 
But it should be pointed out that the lens cannot possibly ' know ' 
what is on the slide; all it can do is to rearrange the light that has been 
scattered during its transit through the slide. Once again therefore 
we see quite clearly that the information about the slide must have been 
on the screen in some form or other all the time. 

Technically the pattern on the screen with no lens in the projector 
is called a hologram — though this word is reserved by some authors 
for cases in which the illumination is rather special (see section 4.3). 
The term simply means that each point on the screen or on a photo- 
graphic plate is receiving information from every point on the object 
(see Fig. 1.1 (a)). 






PHYSICS OF IMAGE-FORMATION 3 

We can demonstrate experimentally that this is so with a simple 
slide — a good subject would be a thick black cross — still with no lens 
in the projector. Place the projector as far away from the screen as 
is possible and then use a converging lens of longish focal length 
(say about 500 mm, though a shorter one may be necessary if the 
projector-screen distance is not large) to form a reduced image of the 
slide on the screen. You will find that a reduced image of the whole 
slide can be produced with the lens at any position within the patch 
of light (fig. 1.1 (b)). This reinforces the assertion that all points on 
the screen contain information about all points on the slide. 

Thus, the first broad stage in the process of image formation may 
be described as the scattering of the incident radiation by the object. 
It could be regarded as a process of coding, and, in order to interpret 
the results, we need a decoding process which in the simple systems 
we have so far considered merely involves the lens. 

I must now digress slightly to draw attention to an important alter- 
native way of seeing. When the lights were out in my first ' silly ' 
experiment, one piece of information that the lecturer might have 
picked up, even in the dark, could have been that someone was smoking: 
in other words he might have seen the glow of a cigarette. Here the 
object is self-luminous and is producing its own radiation. We shall 
see later however that the subsequent process of forming an image 
from the radiation pattern of the self-luminous object is almost identical 
with that of forming an image from the scattering pattern of a non- 
luminous object. (It is in fact exactly the same if the light being 
scattered is incoherent but the process is significantly different from the 
scattering of coherent light.) 

What does the lens need to do ? Fig. 1.1 (a) can help in the explana- 
tion; the lens must take all the elements of information which pertain 
to, say, point A on the slide which would otherwise have been dispersed 
to points P, Q, etc. on the screen and put them together at a single 
point on the screen which will then become the image of point A. 
This is a very remarkable operation indeed but, because it is so familiar, 
and because we can draw neat ray diagrams showing how it occurs, 
we tend to take it for granted. Unfortunately the simple ray diagrams 
of geometrical optics hide a great deal of the complexity of the operation 
and, while telling us where and how big the image is, do not, for 
example, draw our attention to the important problems of contrast, 
brightness and sharpness of the image. 

Thus the second stage of image formation is the recombination of 
the scattered or radiated waves to form an image. Later we shall be 
considering nomenclature in more detail but perhaps it is appropriate 



IMAGES 




(b) 

Fig. 1.1. (a) The hologram relationship: each point of the slide contributes 
scattered light to each point of the screen. (/;) The hologram relationship: 
wherever the lens is placed a complete image of the slide is produced. 



PHYSICS OF IMAGF-IORMATION 5 

to comment at this point that topics such as X-ray, electron or neutron 
diffraction will come within the general purview of our discussions 
just as much as the scattering of light, radio-waves, etc. 

We have thus reduced the image-forming process to two portions, 
scattering, or radiation, and recombination — but we have said nothing 
specific about the operation known as focusing. It is a very familiar 
one; most people can perform it almost instinctively. It is neverthe- 
less worth considerable attention. When one focuses the image of the 
slide on the screen what is the real essence of the operation ? A few 
moments' consideration will lead to the conclusion that all we are 
really doing is to make the image look as nearly as possible like we 
think it should look! If for example, the slide has printing or lines 
on it, we assume that they have sharp edges (fig. 1.2(a)); a slide 
deliberately prepared out-of-focus (fig. 1.2(6)), or a piece of fluted 
glass which has no sharp definition, provides a much more difficult 
problem for the operator (fig. 1 .2 (c)-(f)). In fact there are only two 
possible ways of focusing; we may measure the various distances 
involved and calculate where to put the lens in relation to the other 
components; or we may, by trial and error, make the image resemble 
as closely as possible our notion of the object. It is important to 
realize that these are the only alternatives and we shall see later how 
they influence various applications. 

One practical way of trying to surmount the problem is to focus on 
some part of the object that we recognize and then assume that the 



(a) 



images 



Fig. 1.2. (cont. overleaf). 



IMAGES 



PHYSICS OF IMAGE-FORMATION 



(b) 



teftgr* 







Fig. 1.2. (a) Object with sharp edges in focus, (b) Object with sharp edges 
out of focus. (c)-(/) Fluted glass with four different positions of focusing 
lens; in (d) the chip is in focus. 



rest must then be in focus — and this is the basis of one of the powerful 
techniques in X-ray diffraction (pp. 96-99). Fig. 1.2(d) illustrates 
this point. Focusing is an operation which is always taken very much 
for granted ; it is in reality of such great significance in understanding 
many of the processes of image formation that we shall need to consider 
it later in much more detail. We can summarize the argument so far 
by saying that the process of seeing or imaging involves two basic 
elements — scattering or radiation, and recombination — with the vital 
operation of focusing linking the two. We should now consider the 
limitations that arise before developing the ideas in greater detail. 

1 .2. What are the limitations which restrict the precision of an image ? 

In order to investigate the principal sources of limitation in image 
formation it is convenient to continue our consideration of the experi- 
ment in which the slide was placed in the projector without a lens. 
But the question we now ask is " In what form is the information 
present on the screen? ". The amplitude, or intensity, is more or less 
uniform across the screen and hence the only other variable possible 
would seem to be that of phase. In this example, however, the light 
consists of many separate little trains of waves originating from 
different parts of the source and having many different wavelengths. 
For any one of these trains there will be phase relationships arising 
as a result of the scattering by the object and these lead to some kind 
of interference pattern on the screen. Because the trains are arriving 
randomly, both in terms of their wavelength and of their direction, 
all the interference patterns will be different and the change from one 
to another will occur at such a high rate that we see no effect and the 
screen appears uniformly bright. (This is a manifestation of the 
phenomenon that we call ' incoherence ' which will be discussed at 
greater length in Chapter 2. Under coherent conditions it is possible 
to record the phase pattern and the result is the technique known as 
holography which is described in Chapters 4 and 6.) 

Let us first consider the purely geometrical problems of the intro- 
duction of phase relationships during the scattering process. Here I 
shall simplify matters by considering the effect of just one of the wave 
trains — in this case a plane wave — arriving simultaneously at two 
particular points on the object. A simple experiment with striped 
string or with the Nuffield O-level wave form apparatus is a convenient 
introduction (see fig. 1.3). The string is supposed to represent the 
paths of waves, scattered from two points on an object, arriving at the 
screen, and the stripes — which indicate schematically the crests and 
troughs — help us to see what relative changes in phase will occur. 



8 



IMAGES 



In fig. 1.3 (a) the scattering points are a long way apart and we can see 
that the distance on the screen between points at which the two 
scattered waves are in phase or are in opposite phase (i.e. 180° out of 
phase) is relatively small; in other words there is a pretty rapid alterna- 
tion of phase difference as we move the point of observation across the 
screen. If the points are closer together as in (b) we have to move 
further for a corresponding change in relative phase and, if the points 
are very close together as in (c) we have to move a very considerable 
distance before any significant phase difference is produced at all. 
This kind of experiment allows two points to be made quite clearly. 




Fig. 1.3. Phase relationships for waves scattered from two points in which 
the black stripes indicate crests and the white stripes troughs, (a) Scattering 
points far apart. (/>) Scattering points closer together, (c) Scattering points 
less than one wavelength apart. From The Physics of Musical Sounds, by 
C. A. Taylor, by permission of Hodder & Stoughton Educational. 

First, unless the object is large compared with the wavelength, scattering 
over a fairly wide angular range is necessary in order to obtain signifi- 
cant changes in relative phase and hence significant coding of infor- 
mation. Secondly, if the object is smaller than the wavelength 
(roughly), no significant relative phase change can be made at all, and 



PHYSICS OF IMAGE-FORMATION 



hence it will be impossible to form an image which will contain any 
useful information by any system. This is a simple approach to the 
idea of limit of resolution but is quite valid. 

Thus the first limitation on the precision of the image is imposed by 
the relationship between the wavelength of the radiation and the size 
of detail to be recorded. The second limitation is quite closely related 
and can be illustrated by the same experiment. If scattering occurs 
over a wide range of angles, it may well be that much of the scattered 
radiation will miss the recombination system. The effect of taking in 
only a small cone of radiation with little relative phase change across 
it will be much the same as that of using loo large a wavelength. Thus 
the aperture of the recombining system is the second limitation. 

The third fundamental limitation depends on the nature of the 
scattering process itself. We shall take just three examples. If we 
are using visible light it will be difficult to image transparent objects 
since the interaction between the object and the incident radiation 
will not be very great; with X-rays it will be impossible to record 
any information about the nucleus of an atom because the X-rays are 
all scattered by the electrons and do not interact with the nucleus; 
with microwaves quite different results occur if the scatterers are made 
of dielectric rather than of conducting material even if the objects 
have the same size and shape. 

A fourth fundamental limitation concerns the nature of the inter- 
vening medium. For example an optical telescope is no use in a fog; 
radio waves of certain frequencies cannot penetrate the ionized layers 
in the upper atmosphere; ultrasonic radiation is damped out very 
quickly indeed in water; ultraviolet radiation will not pass through 
glass whereas visible light will; non-uniformities in the medium will 
introduce distortions in the image, e.g. the ' shimmer ' of distant 
objects on a hot day. 

Limitations of a different kind arise from the way in which the 
radiation is detected. For visible radiation we may use our eyes, 
photographic film, or electronic apparatus such as television cameras 
and other photo-sensitive devices. In all of these examples the 
response is proportional to the intensity of the radiation, which is 
proportional to the square of the amplitude, and no direct record of 
relative phase is made. All these detectors are capable of distinguish- 
ing the effects of differing wavelength, though of course the relationship 
between the colour we see or record photographically and the actual 
wavelength present is not a simple one and involves some interesting 
problems of psychology and physiology which are beyond the scope 
of this book. 



10 



IMAGES 



Waves in the radio region, however, will not affect photographic 
plates, and we need to use dipoles or some other kind of radio antennae. 
These devices respond directly to both the amplitude and the phase of 
the signal, and so we are able to record the patterns in greater detail. 
The contrast between the detection of the shorter wavelength electro- 
magnetic radiations and of waves in the radio region is an interesting 
one: for the former we can record the details of a whole scene simul- 
taneously but only in terms of the square of the amplitude; for the 
latter, on the other hand, we can record both the amplitude and the 
phase but only at one point for each detector at a given moment. 

The main limitation imposed by all detectors is that of sensitivity. 
There was a time when the eye was the most sensitive detector of 
visible radiation, but it is now possible to do better with both photo- 
graphic and electronic devices. 

X-rays, ultraviolet rays and y-rays can be detected by the same 
methods as visible light (except, of course, for the eye) and hence no 
phase information can be recorded; ultrasonic radiation and sound 
waves are detected by microphones which are really instantaneous 
pressure measurers and, since they can distinguish between pressures 
that are higher or lower than atmospheric, phase information can be 
recorded. Again, as with radio waves, each detector can record 
information only at one point at a time. 

1.3. To what kinds of radiation do these principles apply? 

The same general principles apply to any kind of radiation though 
the physical mechanism of the scattering processes may be entirely 
different. The techniques used to recombine the image may also be 
totally dissimilar. It is important, however, to be able to grasp the 
interconnections between these apparently unrelated techniques, as 
we shall see later. 

Fig. 1.4 is a chart which gives the wavelengths corresponding to a 
great many different radiations and a list of common objects of sizes 
comparable with the various wavelengths is also included. Since we 
saw that it is virtually impossible to impress on a wave any information 
about detail less in size than its wavelength, we can see that, in order 
to image an object of the size suggested in the table, a radiation which 
appears lower in any column than the object must be selected. 

In the electromagnetic spectrum the wavelengths range from 
thousands of metres down to fractions of a nanometre (1 nm=10 -9 m) 
but the nature of the physical interactions with objects changes con- 
siderably. Radio waves are scattered by both conductors and non- 
conductors, though in somewhat different ways; radar systems may 



PHYSICS OF IMAGE-FORMATION 



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IMAGES 



use either longer radio waves to give general long-range warnings or 
very short waves to give quite detailed images. Infrared radiation can 
pass through rock crystal but is scattered and absorbed by glass. 
The visible region is the most familiar and its properties need no 
elaboration at this stage. X-rays and y-rays interact quite differently 
and are scattered largely by the electron clouds surrounding each atom 
in a solid. 

Acoustic waves follow yet a further different pattern. Scattering 
for all the longer wavelengths, including all the audio region, is almost 
purely mechanical and depends on the geometry and surface properties 
of the object. At the higher frequencies of ultrasonics much more 
penetration occurs and the interactions are more complex. One can 
make practical use of the refractive index between two media for 
ultrasonics (the ratio of the velocities of sound in the two) since 
reflections occur from boundary layers between media of different 
refractive index. 

Finally various particles such as electrons, protons or neutrons may 
be used for imaging and each will have its own peculiarities quite apart 
from the resolution which depends on its equivalent wavelength. 
Electrons — in the electron microscope — have been very powerful 
explorers of objects beyond the limits of optical microscopes, but it 
is necessary to learn how to interpret the results: it is no longer possible 
merely to look at the resulting picture in a subjective way. Neutrons 
can be used and because of their lack of charge may penetrate through 
the electron shells to the nuclei. We shall see in the final chapter how 
the use of these various radiations works out in practice. 



2. A closer look at radiation, 
scattering and diffraction 

" We all know what light is; but it is not easy to tell what it is." 

Samuel Johnson 
BoswelFs Life of Johnson 
Vol. iii, 12 April 1776 



2.1. Radiation and the idea of coherence 

All imaging processes depend on the interpretation either of radiation 
patterns from sources or of scattering patterns of radiation interacting 
with an object. In both cases the radiation is the vital element and 
we must start our detailed study by looking closely at some of the 
ways in which radiation behaves. 

We shall be talking chiefly about the wave aspects of radiation and 
so it may help to create the right pictures in our minds if we think first 
of waves on the surface of water in a ripple tank. Suppose we use a 
single dipper with a fairly small point to create the waves, and suppose 
further that we drive it sinusoidally so that the point is rising up and 
down at a completely regular frequency for as long a time as we wish. 
The result is an infinite train of circular waves radiating out from the 
point. Fig. 2.1 shows this diagrammatically and also gives a photo- 
graph of actual waves. 

If we choose two points such as A and B which are equidistant from 
the source, the wave is always at the same point of its cycle for each; 
cork floats placed at A and B ride up and down exactly in step with 
each other. The same is true if we choose any other point on the same 
circle as A and B. This relationship is an aspect of the wave pheno- 
menon that we call coherence. In this example in which we relate 
the oscillations at a fixed distance from the source (and, since the 
waves travel at constant speed, this implies that we are considering the 
behaviour of points at identical time intervals from the commencement 
of the oscillation) we are dealing with spatial coherence. It could be 
described as the study of phase relationships between the displacements 
at different points in space at a particular instant of time. 



14 



IMAGES 




(a) 



(/» 



Fig. 2.1. (a) The crests of regular waves diverging from a point : cork floats 
at A and B are always in step and demonstrate spatial coherence; floats at 
A and C follow each other at the same frequency with constant phase 
difference and illustrate temporal coherence, (b) Photograph of corres- 
ponding waves on a ripple tank. 



If, on the other hand, we concentrate on a single point such as A 
and consider the oscillation as a function of time, we shall find, in this 
example, that a cork float placed here rises up and down in a perfectly 
regular way as time goes on ; there are no discontinuities or irregularities 
in the graph of its displacement against time. This is an aspect of 
wave behaviour that we call time or temporal coherence. Since the 
wave is travelling at a constant velocity in this example, we can use 
the expression 

distance = velocity x time 

to substitute distance for time, provided that the distance is measured 
along the direction of travel of the wave. Thus the behaviour of a 
cork float at C follows the behaviour of A with a constant time delay. 
We could thus explore the behaviour of the displacement at A as a 
function of time by studying the behaviour of the displacement at 
all points along a line such as AC at a particular moment of time. 
This trick is valuable experimentally and we shall use it quite frequently, 
but there are complications in certain cases which are not as straight- 
forward as the arrangement used here. For this reason I prefer to 
describe temporal coherence as the study of phase relationships between 
displacements at different instants of time at a particular point in 



RADIATION, SCATTERING AND DIFFRACTION 



15 



space. This also has the advantage of preserving a certain symmetry 
with our earlier description of spatial coherence. 

Suppose now that, instead of using a regular sinusoidal drive on the 
dipper, we operate it in some random way. Circular waves still spread 
out from the dipping point and they still travel at a constant velocity. 
The displacements at points A and B are still in step with each other 
but the graph with time of the displacement at A is no longer a regular 
sine curve. Thus we could say that the spatial coherence remains 
unimpaired but the temporal coherence has disappeared (fig. 2.2). 



\ 3i 


B 


c\ 


J 




V 




/ 






(«) 



(b) 



Fig. 2.2. (a) Irregular waves diverging from a point: A and B still exhibit 
spatial coherence but A and C no longer exhibit temporal coherence 
(/>) Photograph of corresponding waves on a ripple tank. 

Now suppose we introduce a number of different dippers randomly 
positioned, and all driven randomly (fig. 2.3). It is fairly clear that 
both the temporal and spatial coherence will disappear and we end 
up with ' incoherent ' waves. Is it a clear-cut ' black-or-white ' 
situation? In other words must the radiation represented by the 
ripples be either coherent or incoherent, or is there a state in between? 

Consider the temporal coherence first. Suppose, in the example of 
fig. 2.1, that instead of driving the dipper with a pure sinusoidal wave- 
form we feed two frequencies in simultaneously. The combined result 
is the well known ' beat ' effect. If the two frequencies are very close 
together the beat ' envelope ' is very long and so there are longish 
periods when A oscillates regularly. If we again make use of the time- 
distance transformation this means that we can move a fair distance 



16 



IMAGES 




(a) 



(b) 



Fig. 2.3. (a) Irregular waves diverging from randomly distributed points: 
both temporal and spatial coherence have disappeared, (b) Photograph of 
corresponding waves on a ripple tank. 



RADIATION, SCATTERING AND DIFFRACTION 



17 



(«) 






(b) 




towards C from A without a discontinuity in the oscillation. If 
however the frequencies are further apart, the trains of continuous 
waves will be much shorter! It turns out that the more indefinite the 
wavelength (or frequency) the shorter is the train and for practical 
purposes we often describe this quantitative aspect of the temporal 
coherence as the ' coherence length '. Fig. 2.4 illustrates these 
principles. 

Waves, of whatever nature, that have well defined wavelength, are 
often described as monochromatic, though strictly this term should 
apply only to light. Highly monochromatic waves have a long 
coherence length and are very coherent temporally; waves with a 
broad frequency distribution have correspondingly shorter coherence 
lengths and are much less coherent. We have thus introduced the 
ideal of partial temporal coherence. 

Suppose now we go back to fig. 2.3 and add a small aperture through 
which the waves must pass (fig. 2.5). We now find that there are 
phase relationships over a certain region in space. In general the 
smaller the aperture the larger the regions of phase correlation. In 
other words the smaller the aperture the greater the degree of spatial 
coherence. So here again we have an intermediate effect — partial 
spatial coherence. Notice, however, in the actual ripple tank photo- 
graph of fig. 2.5 (b) the amplitude of the waves emerging from the slit 



(c) 



Mwmmmmmmmm 



Ul) 







Fig. 2.4. (a) Two sine waves of almost the same frequency. (/>) The sum of 
the waves in (a): the coherence length is relatively long, (c) Two sine waves 
with a much larger frequency difference, (d) The sum of the waves in (c): 
the coherence length is much shorter than for (b). 



18 



IMAGES 



RADIATION, SCATTERING AND DIFFRACTION 



19 




(«) 



ib) 



Fig. 2.5. (a) Restoration of spatial coherence by interposition of narrow 
slit, (b) Photograph of corresponding waves on a ripple tank. 



is very small; we have gained spatial coherence only at the expense of 
a large reduction in amplitude. As we shall see later, this is a point of 
some importance. 

This has been an entirely non-mathematical way of looking at partial 
coherence but it will serve our purpose quite well. The mathematical 
treatment is considerably beyond the level of this book and only 
becomes important when specific systems have to be designed. We 
shall however return to the topic in a little more detail at the end of 
this chapter. 

2.2. Coherence in practice 

Now let us see how the ideas of coherence apply and consider their 
implications for the various forms of radiation that can be used in 
imaging. 

First let us think about sound waves. A single musical instrument, 
such as a clarinet, will produce relatively coherent waves both tem- 
porally and spatially. A single note of definite pitch will have a 
coherence length that is simply equal to the velocity of sound multiplied 
by the time for which the note persists. But how can we attempt to 
measure the spatial coherence? With the water waves we imagined 
two floats and considered whether they were remaining in a constant 
phase relationship. This can be observed directly in the case of the 
water waves and could also be observed relatively directly for sound; 



1 



microphones could be placed at the relevant positions and the resultant 
oscillograph traces compared. Direct observation is not so easy with 
light and other higher frequency radiations however. An almost 
universal technique is to find out whether interference patterns can be 
observed and the extent to which this is possible gives a measure of 
the degree of coherence. We shall consider this in greater detail in 
section 2.8, but for the moment let us see how the idea works out with 
simple experiments in sound. 

If an instrument is played in a room, the reflections from the walls 
will soon build up an interference pattern which will result in the sound 
varying in loudness from point to point. This effect is easily verified 
if you have not met it before by moving your head slightly from side 
to side in a room in which a high-pitched whistle is sounding. As the 
head moves even a small amount the sound changes in loudness. 
Suppose however that an instrument is played with vibrato. This 
usually means that the frequency is ' wobbling ' slightly; this in turn 
means an uncertainty in the frequency and a reduction of the coherence 
length. The result is that the interference pattern is either destroyed 
or moves about and so one ceases to be able to distinguish it clearly. 
A small source of random noise— such as steam escaping from a 
kettle— would have an extremely short coherence length but could 
still be said to create waves with spatial coherence. A very widely 
distributed source such as an audience clapping would produce little 
coherence either temporally or spatially. 

Ultrasonic sources are nearly always highly coherent both temporally 
and spatially. 

Now let us turn to the electromagnetic spectrum. Radio waves at 
all frequencies are usually highly coherent if produced by a single 
transmitter and aerial system. By its very nature, radio-frequency 
radiation stems from oscillations maintained in electrical circuits and 
these tend to be continuous for long periods of time. 

One of the simplest demonstrations of the coherence of radio 
frequency radiation is the interference effects that occur when an 
aeroplane flies over and a television picture alternately fades and 
brightens. In this example the direct waves from the transmitter and 
those reflected from the plane arrive together at the receiver aerial and 
because the radiation is coherent they will have a specific phase relation- 
ship for a given position of the aeroplane. If the waves arrive in step 
the picture will brighten and if the waves are exactly out of step it will 
fade. Then, as the aeroplane moves, the path difference between the 
two sets of waves changes and so the phase difference changes to give 
the familiar alternating effect. 



20 



IMAGES 



Now let us move up through the frequency range of the electro- 
magnetic spectrum. In the infrared region, and indeed through all 
the higher ranges, the processes that generate the radiation are all 
concerned with separate particles in some way. Atoms change from 
one energy state to another and in so doing emit a quantum of radiation 
— a photon as it is usually called. When an electron that has been 
accelerated under the influence of a high potential collides with a 
target X-rays are emitted — but each electron colliding initiates a little 
packet of waves separately and independently of the others. A 
nucleus decays and emits a gamma ray — but again each nucleus in a 
mass of radioactive material behaves more or less independently of 
all the other nuclei. The result is that the radiation emitted lacks both 
temporal and spatial coherence and resembles the water waves of 
fig. 2.3. 

This does not mean, of course, that one cannot have coherent 
radiation in the higher frequency ranges of the electromagnetic spectrum. 
Before 1960, the only way of producing coherent radiation in this 
range was to use the trick which we mentioned in the discussion of 
water waves; if the radiation is sent through an aperture of some kind 
then some degree of spatial coherence will be imposed, and if it is put 
through some kind of filter or monochromator then, the narrower the 
frequency pass band of the filter, the greater will be the temporal 
coherence. The difficulty is that the greater the coherence (of either 
kind) achieved, the more light is thrown away. Thus experiments 
with coherent light were not easy to perform. Alternatively, temporal 
coherence can be achieved if a monochromatic source is used in the 
first place, e.g. in the visible region a sodium flame or lamp gives light 
in which the predominant frequency is about 5xl0 14 Hz and has 
sufficient temporal coherence to make it very useful for interference 
experiments though its brightness is not very high. In 1960, a laser 
was made to operate in the visible region and one of its characteristics 
is that its frequency band may be so narrow that a coherence length of 
tens of kilometres may result! It would be out of place to go into any 
great detail of the mechanism here but we may find it helpful to devote 
half a page or so to the general features of laser radiation which has 
revolutionized experimental optics. 

For optical experiments we find that solid-state lasers are not as 
generally useful as gas-phase lasers. The ruby laser emits single 
flashes of immense power which find all kinds of applications in other 
fields, but the helium-neon or argon-ion gas lasers emit continuous 
radiation which comes very close to the kind of radiation emitted by a 
radio transmitter, from the coherence point of view. Crudely speaking, 



RADIATION, SCATTERING AND DIFFRACTION 



21 






a gas laser consists of a glass tube in which an electrical glow discharge 
is taking place, so that, because of the collisions occurring in the 
discharge, a great many of the gaseous atoms are in an excited state. 
Normally they would revert to the unexcited state randomly and emit 
quanta of light waves in so doing. In the laser system a pair of 
strongly (but not completely) reflecting mirrors is placed, one at each 
end, so that light from within the tube is reflected back and forth many 
times. The mirrors are specially coated so that they reflect strongly 
only one of the characteristic spectrum wavelengths from the discharge, 
and the remarkable effect is that, as the beam travels up and down the 
tube, it triggers off the excited atoms and they emit their quanta in 
step with each other. If the laser is carefully designed it can be made 
to emit a so-called ' uni-phase ' wave front — that is the radiation is not 
only highly coherent temporally but is also spatially coherent across 
the whole width of the beam; the resulting radiation emerges in a 
highly parallel beam without the need for any optical system. The 
light energy is so coherent that interference effects can be produced by 
merely placing two slits in the beam without any source slit. Two 
separate lasers can be made to interfere and produce ' beats ' just as 
two tuning forks can in the acoustic case. Very strict control of 
temperatures and voltages is needed to make this trick work because 
the precise frequency at which the system ' lases ' depends on a variety 
of factors. The basic frequency of a helium-neon laser is about 
4-7 x 10 14 Hz and it follows that, to produce an audio beat stable to 
±10 Hz, the frequency of each laser must be stable to better than 
1 part in 10 13 . High precision indeed! 



2.3. Coherence, bandwidth and the Uncertainly Principle 

Before going much further with this discussion, the somewhat difficult 
concept of bandwidth will need to be elaborated. I propose to make 
this a starred section and it could be omitted, certainly at first reading. 



* Whenever we use radiation of any kind for transmitting information 
from one point to another — whether the information is morse code, 
speech, music, scientific data from space probes, or television pictures — 
the very fact that information is being sent imposes changes in the nature 
of the waves. Let us consider the simple example of using radio waves 
of frequency f c (the so-called carrier frequency) to convey a continuous 
pure musical tone of frequency f m (the so-called modulation frequency). 
Fig. 2.6 shows diagrammatically how this is done; (a) shows the original 
carrier wave, (b) shows the information to be transmitted made everywhere 
positive by adding a constant and (c) shows (a) multiplied by (b). the 



22 



IMAGES 



(a) 



(b) 



(c) 




Fig. 2.6. (a) Trace of carrier wave. (/>) Trace of modulating wave. 
(c) Trace of modulated wave. 

actual modulated wave which goes from transmitter to receiver. Mathe- 
matically we can represent (a) by the equation y e =A$ia2irf c t, and 
(b) by the equation v m = B+B sin 2irf m t, 

The modulated wave (c) is then the product 

v c xy m = [^ sin 2w/c'l [ fi : B sin Mm'], 
which we can rewrite, using standard trigonometrical formulae, as 

y c xy m = AB sin 2 "/c' 'M^ cos 2tt(/ c -/«)' - \AB cos 2tt(/ c +/J/. 

The important point to notice is that we now have the two frequencies 
(/c /m) and (fe-fm) present besides the basic carrier frequency / c . The 
same argument could be used for any kind of information and if we had 
many different modulation frequencies— as for example in the transmission 
of music — many frequencies would be present in the modulated wave. 
If F m is the maximum frequency introduced in the modulation, then all 
these additional frequencies are contained in the band/ c -F m to/ c I F m 
and hence the bandwidth required to transmit this information is 2 F m . 
The consequences of this principle are very wide-ranging. It is, for 



RADIATION, SCATTERING AND DIFFRACTION 



23 



example, one of the reasons why television pictures have to be transmitted 
in much higher carrier frequency ranges than speech or music. The 
picture is built up of 625 lines in each frame, and frames are produced 
at the rate of 25 per second. If we assume that we need detail along each 
line on about the same scale as the vertical detail from line to line the 
total number of changes in information per second must be at least 
625 x 625 x 25 which is approximately equal to 10'. Thus a band width 
of the order of 20 MHz is needed. The medium wave radio band covers 
approximately the range 0-3 to 3 MHz and therefore would clearly be 
useless for conveying T.V. signals as one could hardly imagine modulating 
a carrier at a higher frequency than its own! Nevertheless, for speech 
with a maximum frequency of, say, 10 000 Hz (bandwidth 002 MHz) 
there is no problem. The U.H.F. band (300-3000 MHz) is perfectly able 
to accommodate T.V. bandwidths however. You will notice that, in 
both cases, the bandwidth is about 1 per cent of the carrier frequency in 
the upper part of the frequency range. 

We must now consider how the concept of bandwidth relates to that 
of temporal coherence. You will recall that earlier in the chapter, when 
we considered coherence of water waves, I said that the more indefinite 
the wavelength or frequency, the shorter was the wave train, and we intro- 
duced the term ' coherence length ' as a measure of temporal coherence. 
Now consider a wave which is modulated at a particular frequency. 
Fig. 2.6 (c) shows that the carrier is divided up into groups and that there 
are a number of carrier cycles in each group. Between each group there 
is a region of uncertainty in phase and so the length of each group is a 
measure of the coherence length for the modulated wave. Now the 
number of carrier cycles in the group increases as the modulation frequency 
decreases and so the coherence length is inversely proportional to the 
modulation frequency, i.e. to the bandwidth. 

In round figures, if the bandwidth is x per cent of the carrier frequency 
then there will be of the order of \/x carrier wavelengths per modulation 
cycle. In other words the coherence length expressed in carrier wave- 
lengths is approximately the inverse of the bandwidth expressed as a 
percentage. 

How does this work out for T.V. and radio? In both cases we said 
that the bandwidth is of the order of 1 per cent and so the coherence length 
is about 100 wavelengths. For T.V. this means about 15 m and, for the 
medium wave radio, about 15 km. In the visible region the familiar 
sodium flame has a bandwidth which is of the order of 01 per cent and 
so the coherence length must be of the order of 1000 wavelengths, which 
is approximately 100 X 6x 10 -7 m=6x 10~ 4 m, i.e. 0-6 mm. A helium- 
neon laser on the other hand might have a bandwidth of the order of 
10~ 8 per cent and this leads to a coherence length of the order of 6 km! 

Before leaving this topic it is probably worth pointing out the connec- 
tion between the ideas of coherence and bandwidth and that great funda- 
mental concept of physics known as the Uncertainty Principle, or 
Heisenberg's Principle. Crudely put, it states that the more you know 
of one aspect of a particle the less you know of another. Thus if you 
know precisely where an electron is you can say little about its momentum, 
whereas if you know its momentum precisely you cannot know where it is. 



24 IMAGES 

We have been talking about electromagnetic waves and, as you know, 
the complete mathematical description of their behaviour incorporates 
some elements of wave-like behaviour and some of particle-like behaviour 
and neither is a complete description of its own. For the purposes of 
this book we talk mainly in terms of waves, but the other aspect must not 
be forgotten. In terms of the discussion of the last few paragraphs, if we 
had a wave of completely precise frequency — i.e. a plain carrier with no 
modulation — it would have zero bandwidth and hence infinite coherence 
length. In photon terms therefore its location would be completely 
indefinite. On the other hand a wave of very imprecise frequency, and 
hence large bandwidth, would have a very short coherence length and 
could be located much more precisely in space. The momentum of a 
photon according to the de Broglie relationship is life, where /; is Planck's 
constant, c the velocity of the wave and /'the frequency. So the statement 
of Heisenberg's Principle in relation to electrons can be seen to be 
equivalent to that for photons, with momentum substituted for frequency. 

2.4. The nature of the scattering process 

We saw in Chapter one, in the experiment with a slide in the projector 
but no projection lens, that the patch of light produced on the screen 
must contain, in the form of phase relationships, all the information 
necessary to image the slide. We also saw that each point on the 
screen is receiving light— and therefore information — from each point 
on the slide. Following on from the discussions of the first few sections 
of the present chapter, we should now be able to understand more 
clearly how the lack of both temporal and spatial coherence in the 
projector illumination leads to the lack of any visible interference effects 
on the screen. The phase relationships are there, however, and the fact 
that we can form an image with a lens proves the point. However, for 
the purposes of our present argument, we can consider the source of 
light in the projector as being made up of an enormous number of 
independent point sources each of which is smalt, precisely located, 
and temporally coherent. The whole source is made up of point 
sources of all the different wavelengths present in the white light. 
Each of these separate sources gives a perfectly good interference 
pattern but each pattern is at a different position on the screen and will 
be phase-independent of all the others. Hence they add up to give 
the familiar patch of white light. It will thus be much more profitable 
for us to consider the pattern produced by a single temporally and 
spatially coherent source and, if need be, we can consider problems 
relating to incoherent illumination by adding up the independent 
effects at a later stage (see p. 54). 

Let us therefore consider the geometry of scattering by an object 
placed in a temporally and spatially coherent beam of light — which, 
to make it easy to begin with, we will consider to have plane wave 






RADIATION, SCATTERING AND DIFFRACTION 



25 



fronts. In simpler language it is in a parallel beam of monochromatic 
light! Fig. 2.7 shows the patches produced by two different objects 
in spatially coherent monochromatic light. The first object is regular 
and gives a regular interference pattern which would commonly be 
called a diffraction pattern. The second object is random and gives a 
much less regular pattern which might more commonly be described 
as a scattering pattern. Either term is in fact acceptable at this stage, 
but we shall discuss nomenclature in more detail in the next section. 

We have now reached an awkward point in the development of our 
subject at which a number of ideas are needed simultaneously and yet 
clearly they must be developed in sequence. I shall have to ask you 
to accept one or two statements at face value here and to be patient 
until the ideas behind them can be developed logically later in the book. 
The first is simply to note that we have slipped into the habit of talking 
about diffraction or scattering with visible light. It is so much easier 
to talk about things that we can see directly and many of our discussions 
will be conducted using light as the example, but it is important to 
remember that all the geometrical aspects apply equally well to all kinds 
of radiation and this may not always be stressed explicitly. The 
second concerns the detection or recording of radiation; photographic 
film (or indeed the eye if we are making direct observation) responds to 
the intensity of the radiation (that is to the square of the amplitude) and 
we can make no record of the relative phases. However, when 
scattering patterns are superimposed the phase is of importance. All 
the optical diffraction patterns used in this book are thus records of the 
distribution of the square of the amplitude of the diffracted radiation. 
Such patterns are often referred to in the literature as ' optical trans- 
forms \ The two-dimensional objects containing apertures used to 
scatter or diffract light are usually called ' masks '. 

The third point concerns the way in which we need to look at the 
patterns. Figs. 2.7 (a) and (d) are enlarged prints from masks made up 
of holes which in the original, are 1 mm in diameter. Figs. 2.7 (b) 
and (e) are from strongly exposed negatives, and the central disc 
surrounded by concentric rings in each is related to the size of hole (the 
broad features are in fact those of the Airy disc pattern of any one 
hole, as we shall see later). However, at this point we are only inter- 
ested in the detail arising from the way in which the holes are distrib- 
uted; for this reason we can ignore the rings and study only the detail 
in the central disc. Figs. 2.7 (c) and (/) are exposed and enlarged 
to show mainly the detail in this disc and most of the later pictures 
will be presented in this form. 

Now we can return to our main theme. Whatever the object, it is 



26 



IMAGES 



(a) 



(c) 




RADIATION, SCATTERING AND DIFFRACTION 



27 



(b) 



(«) 



(0 




(b) 



(d) 



(/) 



Ul) 



• # 

• •• 

• 




* - I 



(e) 



Fig. 2.8. (a) Two holes selected from fig. 2.7 (a), (b) Diffraction pattern 
of (a), (c) Another pair of holes selected from fig. 2.7 (a) which are twice as 
far apart as those for (a), (d) Diffraction pattern of (c); the fringes are half 
as far apart as those in (b). 



Fig. 2.7. (a) Enlarged print from mask of holes each I mm in diameter. 
(/>) Strongly exposed diffraction pattern of (a), (c) Diffraction pattern of (a) 
with smaller exposure and greater enlargement than for (b). showing the 
detail within the central disc which is of most interest for our purposes. 
(ct), (e) and (/') as (a), (b) and (c) with a less regular arrangement of holes. 




Fig. 2.9. (a) Amplitude along a line perpendicular to the fringes of fig. 
2.8 (6). (b) Intensity corresponding to (a), (c) Amplitude along a line 
perpendicular to the fringes of fig. 2.8 (d). (d) Intensity corresponding to (c). 



28 



IMAGES 






RADIATION, SCATTERING AND DIFFRACTION 



29 



(«) 




(b) 



<*) 




(0 



(0 



• •••• 
•men 

••••••••• 

• •••• 



(iw) 



(<0 



• •• 




00 



<*) 



(0 




(/) 



(0) 



• •• 



(</) 



(A) 



(J) 



(/) 



(") 




GO 




(r) 




(0 



Fig. 2.10 (co///. opposite). 



Fig. 2. 1 0. (a) and (/) Two different pairs of holes, (c) and (g) Two different 
selections of 4 holes, (e) The complete hexagonal pattern, (b), (d), (/), (A) 
and 0") Diffraction pattern of (a), (c), (e), (#), (/) respectively, (k) to (f) As 
for (a) to (j) for a different arrangement of holes. 



30 



IMAGES 



possible to consider it to be made up of large numbers of pairs of 
points. The two objects chosen for fig. 2.7 are particularly easy to 
divide up in this way but the principle still applies to any object what- 
ever. Every pair of points gives rise to a pattern of fringes which vary 
sinusoidally in amplitude. Fig. 2.8 (b) shows the fringes for the single 
pair of points at (a); Fig. 2.8 (d) shows the fringes from a pair of points 
twice as far apart and in a different orientation, as shown at (c). 
Figs. 2.9 (a) and (b) show plots of the amplitude and intensity respec- 
tively along a line perpendicular to the fringes of 2.8 (/>); 2.9 (c) and 
(d) shows the corresponding plot for 2.8 (d). 

Fig. 2.10 shows how these fringes add together as successive pairs 
of holes are added to make the objects used for fig. 2.7. This addition 
should make clear the point made above that the phase is important 
in the addition, though what we record is the square of the result. 
It does not matter which pairs are chosen and we have built up each 
fringe pattern in two different ways to show that the final result is the 
same. It is important to remember that these patterns are in spatially 
and temporally coherent light and so interference occurs between each 
successive set of fringes and the resultant depends on the relative 
phases of the parts being added. 

Figs. 2.1 1 and 2.12 show some more complex examples of interference 
patterns in coherent light resulting from diffraction or scattering by 
masks which in 2.1 1 can be readily split up into pairs of holes but in 
2.12 are more continuous in form. The mathematical process needed 
to predict the patterns from objects such as those of 2.7 and 2.11, 
where discrete pairs can be identified, is that of Fourier summation. 
For 2.12, where more continuous areas are involved, and it is necessary 
to think in terms of pairs of small elements of area, the process is that 
of Fourier integration. Both of these topics are of enormous impor- 
tance in diffraction and image theory and, though any kind of 
mathematical treatment is beyond the scope of this book, some of the 
physical ideas involved will crop up from time to time later on. A more 
general term that embraces both mathematical processes is ' Fourier 
transformation '. 

Before leaving the geometrical aspects of scattering we shall consider 
a slightly more mathematical discussion of the scattering from two 
points. 

* In fig. 2.13, A and B are two equal scattering points, which are 
supposed to be illuminated by a parallel beam of monochromatic light 
along the direction OP which is the perpendicular bisector of AB. Thus 
the two points A and B are illuminated in phase with each other. At the 
point P on a screen the paths AP and BP are equal in length. Hence 



RADIATION, SCATTERING AND DIFFRACTION 31 



(a) 



B 


• • 
• •••• • • 


• •••••€> >o 


• r » itn ^%Y»i«^ • 


1 • •:••• • 

■ • 



(b) 



(<) 



(e) 



is) 




id) 



(/) 



(h) 



Fig. 2.11. (a), (c), (<?), (g) Four symmetrical arrangements of 1 mm holes. 
(6), (</), (/), (h) The corresponding diffraction patterns. 



~ 



32 



IMAGES 



(«) 




(6) 



RADIATION, SCATTERING AND DIFFRACTION 



33 




Fig. 2.13. Diagram showing change in path length for scattering at an angle 
to the line joining two sources A and B. 



(c) 



w 



(*) 




id) 



(/) 



m 



Fig. 2. 1 2. (a), (c), (e), (g) Four apertures of different shape, (b), (</),(/") CO 
The corresponding diffraction patterns. 



the waves from A and from B arrive in phase and the resultant amplitude 
is twice that due to A or B alone. 

Let us now consider what happens at a point Q on the screen where the 
angle QOP is 0. Now the wave from A has a shorter path AQ than the 
wave from B, which is BQ. If we construct an isosceles triangle A'QB' 
with its base passing through O it becomes clear that the effective path 
difference for the waves from A and B to Q is AA' plus BB'. If is small, 
and if OP is very large compared with AB — both of which conditions 
are usually met in practice because the screen is placed a long way 

from the slits— then AA'=BB'= — sin 8 and the total path difference 

is AB sin 0. Thus the phase lag of the wave from B relative to that from 

A, which we will call 28, is -- •: the path difference, i.e. 28 = — AB sin 8. 

A A 

Now let the distance QP = s and the distance OP=£>. Then, again 
if D is very large and is very small, s= D sin 0. 

Let AB, the separation of the scattering points, be cl. Then 



8 = -ABsin0 
n 



-u 



ds, and of course — is a constant for a given experi- 



ment. 

We now need to find the resultant amplitude at Q from two distur- 
bances of equal amplitude but with a phase difference 28 between them. 
Without going into detail of the form of the disturbance in electro- 
magnetic terms, we can assume that it will be a sinusoidal wave of some 
kind. For convenience, and in order to retain symmetry— which turns 
out to be useful later — let us assume that //there had been a scatterer at O 
it would have created a disturbance at Q of the form 

<£ = flsin (2w/f), 
where <f> is a measure of the disturbance at Q at a time r, a is the amplitude 
and /the frequency. Then the disturbances from A and B will be 
4 A = a sin (2vft+S) and 
<£ B = asin(27r/>-8). 
So the sum which we require is 

tf> A -\-<f> u = a sin (2-nft ) cos 8-\-a cos (27j/r ) sin 8 
+ a sin (2tt//) cos 8- a cos (2tj//) sin 8, 
= la sin (Inft) cos 8 = 2<f> cos 8. 



34 



IMAGES 



From our earlier work this can be written 



S A • 6 n = 2<f> cos 



[(*)*]• 



Thus we have two important results which are at the heart of all 
diffraction theory. The first is that, since the product ds occurs on the 
right-hand side, it follows that if we reduce the slit spacing (</), the spacing 
of the points of similar amplitude on the screen (s) will increase and 
vice versa. The second is that the variation in amplitude is of cosine 
form. Thus since any real object can be considered to be made up of 
many pairs of scattering points, any diffraction pattern can be considered 
to be made up of many cosinusoidal fringes. 

Before leaving this discussion it may be useful for future purposes to 
note that the resultant can be found using a geometrical construction 
called the ' phasor diagram '. A phasor has exactly the same combina- 
torial properties as a vector but is a purely constructional device which 
has no physical entity as does a real vector. 

The diagram we should use is as shown in fig. 2.14. The phasors 
representing the amplitudes transmitted by A and B, which are respec- 
tively a phase angle 5 ahead of a hypothetical wave from O, and 8 behind, 
are shown as thick arrows. The resultant obtained by completing the 
' parallelogram of vectors ' is the dotted arrow. In each case it is easy 
to verify that this is a geometrical solution of the mathematical result 
<j> A -f <£ B = 2(^ cos 5 since the magnitudes of <f> s , and <£ 1{ are both equal to 
the magnitude of <f> a . 

The phasor-diagram technique is valuable in solving many diffraction 
and interference problems. For a complete mathematical discussion of 
the concept a text-book on optics should be consulted (e.g. Optics by 
Smith and Thomson). 



(fl) 




(b) 




- » 



Fig. 2.14. (a) and (b) Phasor diagrams illustrating the resultant scattering 
as the phase angle S changes. 



It is perhaps important to remind ourselves that all these geometrical 
considerations apply whatever the radiation; the theoretical formula- 
tion — apart from problems of scale — is the same whether we are 
considering visible light, radar or acoustic waves. There are however 
physical factors that affect the phase relationships and the most obvious 
is one that leads to a change in velocity of the wave. In optics we 
think of this in terms of the refractive index of the medium; the 



RADIATION, SCATTERING AND DIFFRACTION 



35 



refractive index is simply the ratio of the velocity of the waves in free 
space to that in the medium. (The concept applies equally to other 
forms of radiation and we may quite properly talk of the refractive 
index for ultrasonic waves or for radio waves.) Suppose in a simple 
double slit experiment, one of the slits is covered with a thin plate of a 
medium of a refractive index greater than 1. Tf its thickness is / and 
the refractive index n, the additional optical path length of the waves 
going through this slit relative to the other is (n— \)t and so the net 
result is to move the centre of the pattern over to one side to a point 
at which the phase difference due to the geometrical effect discussed in 
the last section compensates for the phase difference due to the plate. 
A little thought shows that the whole pattern moves en bloc. (But 
see also the end of Section 2.8 for a complication that occurs if the 
light is not highly monochromatic.) 

This is a relatively easy problem to deal with, but if the object causing 
the additional path is irregular then the situation can be very difficult. 
When one considers that the wavelength of light is of the order of 
5 x 10 -7 m it is obvious that a change in thickness of a piece of glass 
of refractive index ~ 1-5 of the order of only 0-5 microns would lead 
to a phase difference of 180°. In producing diffraction patterns, 
therefore, it is usual to use slits and holes rather than black and white 
patterns on a photographic plate. The masks used for Figs. 2.1 1 and 
2.12 were produced by making holes in thin, opaque, * estar ' sheet and 
the contact prints are approximately actual size. 

2.5. Nomenclature of diffraction and scattering processes 

The nomenclature of scattering and diffraction processes is highly 
confused, and in order to make progress the only real possibility is to 
define a set of terms that are self consistent — which is more than some 
text books do, I regret to say — and then to stick to them even though 
they may differ from those used by other authorities. I prefer to keep 
the term diffraction to describe any passive interaction between 
radiation and an object; thus in my terminology scattering includes 
diffraction but may also include active interactions (such as for example 
Compton scattering where a change of wavelength occurs). Since 
discussions of these forms of scattering are not included in this book, it 
follows that for our purposes scattering and diffraction are synonymous. 
Interference is the interaction of radiation with itself, i.e. without the 
presence of material objects. Thus, in my terminology, in the familiar 
Young's double-slit interference experiment, scattering or diffraction 
occurs at the first single slit and again at the pairs of slits: the resultant 
wave trains then interfere to produce the pattern. In a Newton's 



36 



IMAGES 



rings experiment the various trains of waves are created by multiple 
reflection and transmission and their subsequent interaction is inter- 
ference. In a diffraction grating the incident radiation is scattered or 
diffracted by the grating and the subsequent wave trains interfere to 
produce the pattern. If this scheme is followed it leads to unambiguous 
descriptions of the various processes. 

My only real regret is the need to use the term ' interference ' for the 
interaction of waves. In linear systems the principle of superposition 
is obeyed — that is the resultant disturbance at any point is the sum of 
all the separate disturbances at that point — and so the waves have no 
permanent effect on each other and so ' non-interference ' would be a 
better term. However, the term has been in use too long to change 
now ! What happens, of course, is that a collection of waves passing 
through a point lead to a certain resultant at that point as viewed by 
an outside observer, but we must remember that even at a null point 
in an interference pattern, waves are continuously passing through and 
conveying energy; it is merely the external combined result that is zero. 
The effect is somewhat like that of a man climbing up an escalator at 
the same speed as it is descending; to an observer some distance away 
who cannot see his legs he appears stationary — though clearly this is 
the result of two movements which neutralize each other as far as the 
outside observer is concerned. 

In classical treatises on diffraction it is customary to divide phenomena 
into two categories which are usually described as ' Fraunhofer ' and 
' Fresnel '; the impression is sometimes created that there are just two 
patterns that any particular object can produce — one in the Fraunhofer 
category and the other in the Fresnel category. There is however an 
infinite number of patterns and the two often described in the literature 
are limiting cases. 

Fraunhofer diffraction is quite specific and we can speak of ' the ' 
Fraunhofer diffraction pattern of an object to describe the pattern 
resulting from subsequent interaction of the waves when radiation is 
scattered by the object under the following conditions: 

(1) The radiation is monochromatic, i.e. is temporally coherent; 

(2) The incident radiation has plane wave fronts and is laterally 
coherent; 

(3) The subsequent interference pattern is viewed at infinity — or in the 
back focal plane of a converging lens receiving the scattered 
radiation, which of course is equivalent to viewing at infinity. 

Under these conditions the only readily observable variations in pattern 
(not counting absolute intensity level — which depends on the intensity 



RADIATION, SCATTERING AND DIFFRACTION 



37 



of the incident radiation) for a given object are of scale and of extent 
and these factors depend on the relative scale of the object and wave- 
length. The cautious phrase ' readily observable ' is inserted because, 
although the statement is absolutely true for the intensity distribution 
in the pattern (which depends on the square of the amplitude), the phase 
distribution depends on other factors which we have not included here. 
However, the phase can be observed only in the special circumstances 
of the addition of a coherent beam as in holography (see Chapter 6) so 
the effect need not worry us unduly. 

Reference was made earlier to the fact that the mathematical relation- 
ship between an object and its scattering or diffraction pattern involves 
Fourier transformation. In more precise terms, if we add to the three 
conditions for Fraunhofer diffraction specified above, a fourth: 

(4) The diffracting object is placed in the front focal plane of the 
converging lens specified in (3), 

then the distribution in its back focal plane (assuming that the lenses 
are fully corrected for aberrations) is exactly described by the Fourier 
transform of the object in terms of both amplitude and phase. I 
suppose, strictly speaking, that this is the only true Fraunhofer diffrac- 
tion pattern. 



Object 



Diffraction 
pot tern 




Diffraction 
pattern 



Fig. 2.15. (a) Simple arrangement for observing Fraunhofer diffraction 
which approximates to conditions 1, 2 and 3 on page 36. (b) Arrangement 
for observing Fraunhofer diffraction that conforms with conditions I, 2 and 3 
on page 36. (c) Arrangement for observing Fraunhofer diffraction that 
also satisfies condition 4 of page 37 and which therefore gives true Fourier 
transforms. 



38 



IMAGES 



However, since we are normally concerned only with intensities, 
particularly if we are taking photographs, this restriction need not be 
applied rigorously. Indeed, the patterns produced under conditions 1, 
2 and 3 above have already been described as optical transforms to 
distinguish them from precise Fourier transforms. The simple set-up 
often used in school laboratories, shown schematically in fig. 2.15 (a) 
is a reasonable approximation to the arrangement shown in fig. 2. 1 5 (b) 
which does conform with conditions 1, 2 and 3. It will certainly give 
Fraunhofer diffraction patterns, or optical transforms, whose intensity 
distribution will be virtually indistinguishable from those produced by 
the arrangement of fig. 2.15 (b), and also from the patterns produced 
by the arrangement of fig. 2.15 (c) which is the arrangement satisfying 
all four conditions, and hence giving correct phases. 

It is convenient to describe all the patterns produced when conditions 
2 and 3 are not obeyed as Fresnel patterns — though there are of course 
an infinite number associated with each object depending on the nature 
of the incident wave fronts and of the viewing system. As an example 
we give in figs. 2.16 (b)-(f) the Fraunhofer diffraction pattern and 
four different Fresnel patterns of the same hexagonal aperture. 

In most of the applications described in this book the object will be 
far enough away from the source of radiation for the wave fronts to 
be nearly parallel and the viewing arrangements will not depart much 
from condition (3) above and so the intensity distribution in the 
relevant pattern will be little different from the Fraunhofer pattern. 
Mathematically speaking we often describe this as the ' far-field 
approximation '. There is a mathematical formulation which applies 
in general but which is greatly simplified when the three conditions are 
fulfilled and the name derives from this. 

Finally it is important to notice that there is often confusion over the 
differences between so-called one, two and three dimensions in relation 
to diffraction. In fact, of course, diffraction can only be thought of in 
three dimensions. The most useful concept in interpreting diffraction 
is that of Huygens' principle and this depends on the idea that each 
part of the wave front acts as the centre of a new spherical wave; this 
immediately means that one must think in three dimensions. We shall 
see later that, whereas X-ray diffraction is often claimed to be essentially 
three-dimensional because one normally thinks of the interaction with 
the atoms in a crystalline solid, optical diffraction is usually thought of 
in terms of the interaction with a planar object. We shall see that the 
significant differences between X-ray and optical diffraction do not 
arise from this point but from the difference in the relative scale of the 
radiation and of the object: X-rays have wavelengths comparable with 



RADIATION, SCATTERING AND DIFFRACTION 



39 




Fig. 2.16. (a) Contact print of hexagonal aperture, (b) Fraunhofer 
diffraction pattern of (a), (c), (d), (e) and (/) Four different Fresnel 
diffraction patterns of (a). The scales of the five patterns have been adjusted 
to make them all comparable in size. 



40 



IMAGES 



the atomic separation; objects used in optical diffraction experiments 
are usually large compared with the wavelength of light. One par- 
ticularly beautiful exception is the gemstone opal. The ever-changing 
colours produced when an opal is viewed from different directions are 
diffracted beams from a three dimensional arrangement of transparent 
equal silica spheres whose diameter is a little smaller than the wave- 
length of light. Fig. 2.17 is a scanning electron micrograph showing 
the spheres. Their refractive index is 1-45 so their optical diameters 
are 1 -45 x 2-4 x 10 -7 m = 3-48 x 10 -7 m. The path difference introduced 
when light is scattered back along its incident path by successive 
spheres is thus 6-96xl0 -7 m. This corresponds to a wavelength in 
the red region of the spectrum and so reinforcement of red would 
occur; at other angles the path difference is less and other colours of 
shorter wavelength can be reinforced. 




Fig. 2.17. Scanning electron micrograph of a fragment of opal at a magnifi- 
cation of 5000. The spheres are of transparent amorphous silica about 
240 nm in diameter. Photo by Mrs. C. Winters, Department of Zoology, 
University College, Cardiff. 

2.6. How can we observe optical diffraction patterns ? 
Most people will have observed diffraction patterns at some time or 
other — though they may not have been conscious of the fact. Look 
out of the window of a bus at a street light when the window is misted 
or spotted with rain and you will see halos round the light; clean the 
window by rubbing your glove horizontally across the glass and you 
see vertical streaks spreading out from the light; look through an 
umbrella at a distant street light and you will see a pattern of spots 



RADIATION, SCATTERING AND DIFFRACTION 



41 



arranged in a regular way. These are all diffraction patterns and 
indeed are very close to being Fraunhofer diffraction patterns as far as 
their intensity distribution goes. The reason is that the light source 
in each case is fairly distant and hence the incident light is almost 
parallel; if the eye is focused on the source, then the image on the 
retina is in the back focal plane of the lens of the eye and so the 
Fraunhofer conditions are very nearly obeyed. 

In my terminology, the familiar Young's double-slit experiment is 
really a diffraction experiment; the slits interact with the wave fronts 
and produce diffracted waves which subsequently interfere to produce 
the fringe pattern. In the laboratory it has been the practice for many 
years to use slits because this enables us to produce brighter patterns. 
Slit sources do however obscure the essentially three-dimensional 
nature of the diffraction process which was mentioned towards the 
end of the last section. Most of the illustrations in this volume will 
therefore be produced with point rather than slit sources but, in order 
to underline the relationships with experiments that may be more 
familiar, fig. 2.18 shows a comparison of the patterns produced by two 
different double-slit spacings and two different slit widths, using both a 
slit source and a point source. 

The simple experiment with a street lamp forms the basis of the 
simplest available method of observing diffraction patterns in the 
laboratory. Set up a simple pea-lamp or flash-lamp bulb at one end 
of the laboratory and view it, with the eye focused on it, through a 
fine handkerchief, a piece of umbrella fabric or a silk scarf. The 
regular pattern of spots is related to the fine structure of the fabric. 
The pattern will, of course, be coloured if white light is used, as there 
will be a pattern on a different scale for every wavelength present; the 
overlap clearly produces a coloured result. A piece of green cellophane 
over the lamp produces sufficiently * monochromatic ' light to permit 
the individual pattern for one wavelength to be isolated fairly easily. 
Replace the regular fabric by a piece of knitted fabric or of a pair of 
' stretch nylon * tights and the regularity disappears and a somewhat 
diffuse halo is seen. This is still a diffraction pattern and is related 
in the same mathematical way to the sub-detail of the fabric. Figs. 
2.19(6) and (/?) show photographs of the diffraction patterns of two 
such pieces of material. 

Stretch the material in the horizontal direction and notice that 
the horizontal dimensions of the pattern shrink (figs. 2.19 {cl) and (/)). 
Then tilt the unstretched fabric so that the light passes through it at 
about 60° to the normal. Now the projection of the fabric along the 
light direction has shrunk to half its original dimensions (cos 60° = \) 

D 



42 



IMAGES 



RADIATION, SCATTERING AND DIFFRACTION 43 



and the details of the diffraction pattern expand to twice the size 
(figs. 2.19 (f) and (/)). 

As a final experiment with this simple arrangement translate one of 
the pieces of fabric from side to side as you observe its diffraction 
pattern. The pattern does not move, although if there are variations 
in the regularity or spacing of the threads the pattern may change 
slightly. These experiments illustrate some of the very important 
properties of diffraction patterns and (see section 2.3) hence of some of 
the corresponding mathematical formulations — Fourier transforms. 

It may be worth considering for a moment the explanation of these 
phenomena. The reciprocal effect — that the pattern shrinks or 
expands when the object expands or shrinks — can be understood most 
easily by referring back to section 1.2 and fig. 1.3. The striped string 
experiment shows that when points in the object are close together one 
has to move further away from the centre line to find the same phase 
difference and vice versa. 

The lack of movement of the pattern when the object is moved is not 
difficult to understand. The image of the lamp is focused at one 
point of the retina and unless the lamp is moved relative to the eye 
this point will always be at the centre of the pattern. All waves 
diffracted in a particular direction in space that are parallel with each 
other will come to a focus on the retina at a fixed point some distance 
from the centre and, since lateral translation of the object moves all 
these waves parallel to themselves the actual pattern does not move. 



(a) 



(b) 




(c) 



V) 



Fig. 2.18 (cont. opposite) 

Fig. 2.18. (a) Two rectangular apertures, (b) Two rectangular apertures 
further apart, (c) Two rectangular apertures with spacings as for (a) but each 
wider, (d) Two rectangular apertures with spacings as for (b) but each wider. 
Opposite: (e) Diffraction pattern of (a) with slit source. (/) Diffraction pattern 
of (a) with point source, (g) Diffraction pattern of (b) with slit source. 
(A) Diffraction pattern of (b) with point source. (/') Diffraction pattern of 
(c) with slit source. (J) Diffraction pattern of (c) with point source. 
(k) Diffraction pattern of (d) with slit source. (/) Diffraction pattern of (d) 
with point source. 



(e) 




(/) 



(*) 




(h) 



0") 



• 




0) 



(*) 




(/) 



44 



IMAGES 



RADIATION, SCATTERING AND DIFFRACTION 45 



ui) 













(b) 



(/) 




0) 



(c) 




i<n 



(A) 




(/) 



(e) 




If) 



(in) 




(") 



(g) 




m 




(p) 



Fig. 2A9 (see page 46). 



Fig 2.19. (see page 46) 



46 



IMAGES 




(a) 



(r) 



Fig. 2.19. (a) and (g) Enlarged view of umbrella fabric and of stretch tights. 
(b) and (/?) Diffraction patterns of (a) and (g) in laser light, (c) and (/') The 
fabrics of (a) and (g) stretched horizontally, (d) and (j) Diffraction patterns 
of (c) and (i) in laser light, (e) and (k) The fabrics of (a) and (g) tilted out of 
the plane of the page. (/) and (/) The diffraction pattern of (e) and (/c) in 
laser light, (m) to (r) are all diffraction patterns of fabric (a): for (to) and (//) 
the source of light is white and is small and large respectively; for (o) and (/?) 
the source is green and is small and large respectively and for (q) and (r) the 
source is a He-Ne laser and the source is small and large respectively. 



The effects of reduced spatial coherence can, of course, be demon- 
strated by enlarging the source and those of increased temporal 
coherence by changing from white to green light. A greatly improved 
way of observing the same phenomena is to place the fabrics in the 
unmodified beam of a helium-neon laser. The patterns will then be 
seen on the screen on which the laser beam ultimately falls. Fig. 2.19 
illustrates these various effects using the umbrella fabric of fig. 2.19 (a) 
as the diffracting object. 

For research purposes a somewhat more sophisticated system is 
necessary and fig. 2.20 (a) shows a diagram of the system used in the 
author's laboratories at Cardiff. Fig. 2.20 (b) is a photograph of the 
actual apparatus. A carefully designed optical system h^ (which 
involves three stages) expands the laser beam, without losing its spatial 
coherence, so that a uniform patch of light falls on lens Li after which 
the light becomes parallel. Between Lj and L 2 we thus have a very 
accurately uniform set of plane parallel wave fronts which will fall on 
any object placed at P. Lens L 2 (in the absence of an object at P) 
produces a minute focal spot of intense brilliance in its back focal 
plane at Q. If an object is placed at P, then this spot (strictly, an Airy- 
disc) becomes the Fraunhofer diffraction pattern of the object. 

For the experiment described in sections 5.1 , 5.2 and 6.6 an additional 
lens Lg may be placed in position in such a way as to produce an image 



RADIATION, SCATTERING AND DIFFRACTION 



47 




to! 




(b) 



Fig. 2.20. (a) Diagram of diffraction system in the Department of Physics 
at University College, Cardiff (the diffractometer). (b) Photograph of the 
system outlined in (a). 



of the object P on a screen R. We now have a complete demonstration 
of the two stages of image formation; from P to Q we have scattering 
and from Q to R we have recombination. With a 50 mW helium-neon 
laser this system is capable of producing, for example, a diffraction 
pattern of a 1 cm diameter hole which has its central disc a centimetre 
or so in diameter on the screen but is quite bright enough to be seen 
with the naked eye in a darkened room. 

Simple objects in the diffractometer give useful demonstrations of 
Fraunhofer diffraction principles and with complex objects extremely 
beautiful patterns can be produced. A selection illustrating different 
points is shown in figs. 2.21 and 2.22. 

This system was developed primarily for use in interpreting X-ray 
diffraction patterns and the applications are discussed in rather more 
detail in sections 4.2 and 6.10. 

2.7. How can we record scattering patterns for non-visible radiations? 

So far we have tended to use optics only for our illustrations. The 
reason is simply that optical patterns are by far the easiest to record. 
A suitable photographic film enables one to record instantly the whole 
two-dimensional distribution of intensity, which depends on the square 
of the amplitude. But we should now examine more closely the fact 



48 



IMAGES 



RADIATION, SCATTERING AND DIFFRACTION 



49 



that it is only a quantity that depends on the amplitude that we can 
record; the phase is not recorded. Indeed our eyes are not sensitive 
to phase, nor is any other system of recording in the ordinary way. 
(The exception is in laser holography — a special case which we discuss 
in Chapter 6.) 

Why can we not record the phase differences? Consider yellow 
light with a wavelength of 5-9 x 10 ' m (sodium light). The corres- 
ponding frequency, if we take the velocity of light to be 3 • 10 8 m s ', 
is about 5 x 10" Hz. In other words a phase difference of half a cycle 



(a) 



B 


.illllllii 


. 


P 

I 

l| 

it. 


• 
llllllll' 


1 



(b) 



(0 



(<') 



• • 


1 


ttrl 


ft i 


SmiJJV 


• • • 


'* 


|i 



(</) 



(/) 



(g) 




(//) 



Fig. 2.21. (a), (/>) Two holes and their diffraction pattern exposed to show 
rings, (c), (d) Two larger holes with the same spacing as (a) and their 
diffraction pattern exposed to show rings, (e), (f) Three holes and their 
diffraction patterns. Note subsidiary maximum. (#), (It) Twelve holes and 
their diffraction patterns. Note subsidiary maxima. 



(«) 



(c) 




d>) 



(d) 



Fig. 2.21 (cont. opposite). 



Fig. 2.22. (a) and (c) Masks chosen to give aesthetically attractive diffraction 
patterns (b) and (d). From An Atlas of Optical Transforms, by G. Harburn, 
C. A. Taylor and T. R. Welberry, by permission of G. Bell & Sons, Ltd. 



50 



IMAGES 



involves a time measurement of 10~ 15 second. This is certainly 
smaller than any time measurement of which our technology is capable 
at the moment. The same is not true of course for other radiations. 
In the case of radio waves, phase differences can be measured as indeed 
they can for sound waves — but of course photographic films cannot 
be used for them. Thus in order to record patterns of radio or sound 
waves we usually move a detector from point to point in the area 
concerned. 



(a) 





(/>) 



(<0 



(</) 



Fig. 2.23. (a) Photograph of " Lloyd's mirror " fringes using a laser source 
illuminating a slit: interference occurs between light emerging from the slit 
and that reflected from a sheet of glass, (b) Plot of intensity variation in 
Lloyd's mirror fringes produced with a microwave source (A=3cm) and a 
wooden bench top as reflector, (c) Plot of intensity variation in Lloyd's 
mirror fringes produced with sound waves (frequency = 11 kHz. A=3cm) 
using the wooden bench top as reflector, (cl) Densitometer trace of the 
photograph of (a). The scales have been adjusted to give comparable sizes — 
the difference in location of the first maximum arises from the different phase 
shifts at the mirror. 



RADIATION, SCATTERING AND DIFFRACTION 



51 



Fig. 2.23 shows a comparison between so-called ' Lloyd's mirror ' 
fringes produced (a) optically, (b) with microwaves and (c) with 
ultrasonics. For comparison purposes we show also at (d) a micro- 
densitometer trace of the pattern of (a); the departure from a cosine 
curve is due to the non-linearity of response of the photographic film. 

Electron patterns may be recorded either with a photographic plate 
or by means of a positively charged collector which detects the arrival 
of negative charges. Other particles may be detected by their ionizing 
properties and indeed in the early days X-rays were detected that way. 
X-rays also affect a photographic plate but more recently the practice 
has swung round again and it is again common to record X-rays with 
Geiger or scintillation counters. 

2.8. How can we measure coherence? 

Now that we have studied the nature of radiation and have seen some- 
thing of the way in which scattering or diffraction patterns can be 
observed or recorded, we can return to the problem hinted at in 
section 2.2, the measurement of coherence. This is a somewhat 
complex operation and for this reason the section has been starred 
and may be omitted at first reading. 

* Let us consider spatial coherence to begin with. Most of us will have 
carried out some kind of double-slit experiment using either a sodium 
flame or a sodium lamp and will be aware that an initial ' source-slit ' is 
necessary between the flame and whatever device is being used to produce 
the two effective sources. We shall assume for the moment that Young's 
arrangement with two further slits is being used as in fig. 2.24, but the 
principles apply equally well to systems using a bi-prism, split-lens, etc. 
What is the process by which the single source slit introduces spatial 
coherence? It is, in fact, itself a diffraction process. 



Double 
slit 



Source Source 

of liqhl sli' 





Amplitude 



Fig. 2.24. Diagram showing how a single slit introduces some degree of 
spatial coherence in illuminating a pair of Young's slits. 



52 IMAGES 

Fig. 2.24 is intended to be a section of the system by a plane perpen- 
dicular both to the planes of the slits and screen and to the length of the 
slits. Consider one point of the source in this plane such as A. If this 
were the only source of illumination the single slit would produce a 
diffraction pattern whose cross-sectional amplitude distribution would be 
as shown at A'. (It would look something like fig. 2.18 without the fine 
fringes. Similarly a point such as B would produce a pattern at B' and 
C at C" etc. When the whole source is considered, the illumination 
of the double slit will be the superposition of a great many laterally 
displaced single-slit patterns, and for many of these the two elements of 
the double slit will both be within the same central peak. This is the 
origin of the coherence introduced by the single slit. 

Now if the source-slit is broad, the diffraction pattern will have a 
narrow central peak and so the chance of many such peaks lying over the 
two slits is small and the coherence is low. If the source slit is narrow, 
its diffraction pattern has a broad central peak and many such peaks may 
span the two elements of the double slit and the coherence is high. 

Notice that for certain single slit widths it is possible for the central 
peak to be over one element of the double slit and the first subsidiary 
maximum — which is in the opposite phase — to lie over the other. The 
number of elements of the source for which this would occur is small but 
nevertheless it does suggest that negative coherence might be possible. 
We shall return to this point a little later. 

Now we must consider the experimental measurement of coherence. 
If the illumination of the double slit were totally coherent spatially, that 
is, if we had parallel wave fronts impinging on it, and if the two elements 
of the double slit transmitted identical amplitudes, then the interference 
pattern produced would have an intensity distribution which would 
vary sinusoidally. It would alternate from some maximum value to 
zero. If the light were totally incoherent spatially, then the sinusoidal 
distribution would disappear. It should not be too surprising to find, 
therefore, that for partially coherent light we have a mixture of the two — 
that is the elements of the source giving central peaks covering both slits 
give a sinusoidal distribution with real zeros and the elements not covering 
both slits give uniform illumination. The more highly coherent the light 
illuminating the two slits the nearer will the low points in the sinusoidal 
approach to zero and the contrast or * visibility * of the fringes is a measure 
of the degree of coherence. 

In mathematical terms the visibility function V is defined as 
(Anax-'min'/max-'min), where /„,„ is the intensity of the central bright 
fringe and / min that of the first minimum, and this quantity Khas the same 
magnitude as the degree of coherence. Fig. 2.25 shows three patterns 
using the same pair of double slits for each. For (a) the source is very 
small and therefore the coherence is high; for (b) the source is larger, the 
coherence much lower, and in fact, has the negative correlation referred 
to above, as witnessed by the intensity minimum at the centre; for (c) it is 
larger still and the coherence is lower, but the correlation is again positive. 

What about the measurement of temporal coherence? You will recall 
that in section 2.1 temporal coherence was described as being concerned 
with the phase relationships between displacements at different instants of 



RADIATION, SCATTERING AND DIFFRACTION 



53 



ooo 





(«) 



m 



(O 



Fig. 2.25. (a) Diffraction pattern of two holes with, below, the theoretical 
intensity curve, (b) As for (a) but with illumination of lower coherence and 
showing negative correlation, (c) As for (a) but with illumination of even 
lower coherence showing restoration of positive correlation. By permission 
of Professor B. J. Thompson, Institute of Optics, Rochester, N.Y. 




Fig. 2.26. Two apertures with a glass wedge over one which when placed 
in the parallel beam section of a diffractometer (e.g. at P in fig. 2.20 (a)) 
permits the measurement of temporal coherence. 



54 



IMAGES 



time at a particular point in space. We also saw that the length of time 
for which a wave passed a particular point without a discontinuity — 
which is the measure of temporal coherence — could be translated into a 
coherence length simply by multiplying the time by the speed of travel 
of the waves. This coherence length forms the basis of methods of 
measuring temporal coherence which, in fact, closely resembles the 
technique for measuring spatial coherence. Fig. 2.26 shows a suitable 
experimental arrangement in principle. It is a two-slit diffraction 
experiment in which an additional optical path length can be introduced 
for the waves passing through one of the apertures. Ideally the apertures 
should be small holes and a wedge of glass or quartz is placed over one of 
them. By sliding the wedge to and fro, the additional optical path, 
introduced because of the reduction of the velocity of light in the wedge, 
can be increased until the visibility of the fringes, produced when the 
emergent waves interfere, disappears. The additional optical path is 
then a measure of the coherence length. Since one path is in air and the 
other in a medium of refractive index n (i.e. the velocity is reduced in the 
medium in the ratio // : 1) the additional path is (n-l)t, where / is the 
thickness of the wedge at the position of the hole. 

2.9. Images with self-luminous and incoherently illuminated objects 
In most of the detailed discussions of this book we confine attention to 
coherent illumination, but it is worthwhile to consider briefly the 
consequences of departure from coherence. If we consider scattering 
from any two points on the object, the resultant pattern is of sinusoidal 
fringes (see, for example, fig. 2.8). If the illumination is incoherent, 
or the two points are self-luminous, the pattern is still of sinusoidal 
fringes instantaneously, but the positions of the fringes continually 
changes as the relative phase of the waves radiated from the two points 
changes. The net result appears to be uniform illumination. 

However, if we then proceed to the second stage of imaging — 
recombination — the instantaneous phase relationships are preserved 
and the image is formed without difficulty, regardless of the illumination. 
Thus for any process that involves recording of a scattering pattern, 
interpretation will only be possible with coherent illumination. But if 
we continue with recombination to produce an image the process is 
still possible whatever the illumination. 

One final, and important, point is that (as we shall see in Chapter 5), 
in all practical imaging systems, the scattering pattern is limited, or 
modified, by apertures or defects with resulting degradations of the 
image. Detailed studies, that are beyond the scope of this book, 
show that the kind of degradation depends on the nature of the illumi- 
nation and incoherent illumination can result in better resolution. 
(A hint towards an explanation of this is that, statistically, the rapid 
phase changes lead to more information entering the recombination 
stage). 



3. Principles of direct 

recombination processes 



" all concentrating, like rays 

Into one focus kindled from above; " 



Lord Byron 

Don Juan, canto ii st 186 



3.1. ' Straight line'' imaging with pinhole cameras 

We must now return to the problem of how the second stage of the 
image-forming process can be achieved. Let us think back to the 
experiment in which we placed a slide in a projector with no projection 
lens and produced a patch of light on the screen: is there an even 
simpler way of producing an image than by replacing the lens? We 
said that the problem is just that every point on the screen receives 
information from every point on the slide and we are really being 
embarrassed by an excess of overlapping information. Is there any 
way in which we can, by sacrificing a great deal of the information, 
still leave ourselves with enough in a more manageable form to enable 
us to form an image? The answer is surely 'yes'. By placing a 
card between the slide and the screen and piercing one small hole in 
it we can ensure that light from each point on the slide falls on only a 
very small region of the screen and simple straight line geometry shows 
us that the arrangement of the patches on the screen will be exactly 
the same as that of the corresponding areas on the slide, except for 
inversion of the top and bottom and left and right (fig. 3.1). This, 
of course, is the principle of the pinhole camera. The size of the 
image is determined purely by the geometry and depends only on the 
distances between the components (fig. 3.2). 

This system is so simple that it may be at first surprising that it is 
not more widely used. What are the snags? The first is that, if the 
hole used is large enough to produce a bright image, then the patches 
of information overlap and the result is blurred: if, on the other hand, 
the hole is made smaller then a sharper image results but, since more 
of the light is being cut out by the screen, the image is very dim. The 



56 



IMAGES 




Fig. 3.1. Principle of the pinhole camera. 




Fig. 3.2. The size of ihe image is determined by the ratio of the distances 
between components. 



(Opposite). Fig. 3.3. (a) and (e) Objects photographed with a good lens. 
(b) and (/) Images with a large pinhole, (c) and (g) Images with a medium 
pinhole, (d) and (h) Images with a small pinhole. 



DIRECT RECOMBINATION PROCESSES 



57 



(«) 



(c) 



ti\ 



I 



(e) 



(*) 





(b) 




UO 




(/) 




(/') 



58 



IMAGES 




(a) 




(b) 




(c) 
Fig. 3.4. (cont'._opposite). 



DIRECT RECOMBINATION PROCESSES 



59 




(d) 

Fig. 3.4. (a), (b) and (c) X-ray pinhole photographs of target of X-ray tube 
showing progressive deterioration. (</) Photograph of target after tube had 
ceased to operate and had been dismantled. By permission of Mr. W. 
Sutherland, Velindre Hospital, Cardiff. 



Oblique 
lorqet 



Water cooling 
/pipes 




Electron 
gun 



Thin metol 
"window" 



Lead 
pinhole 



Light shield' 




(/» 




Fig. 3.5. (a) Experimental arrangement for producing fig. 3.4 (a)-(c). 
(h) Cross-section of lead ' pin-hole ' used for fig. 3.4 («)-(<■). 



60 



IMAGES 



second is that if the hole is made really small in an attempt to obtain 
very sharp images the desired result does not follow. Apart from the 
fact that the image may become too dim to see, the hole produces its 
own diffraction pattern at each point of the image and, as the hole gets 
smaller, its diffraction pattern gets larger; the patches thus begin to 
spread again. In the limit when the hole becomes smaller than the 
wavelength of light the light emerges from the hole and is scattered 
uniformly in all directions and so we are back to the first stage of total 
overlap of information and with hardly any light anyway! 

Fig. 3.3. shows pinhole photographs of two sorts of object— one 
made up of separate points, which enables us to see the effect of the 
pinhole size and the other a rather more complicated— and attractive 
—object; images produced with three sizes of pinhole are given together 
with a photograph using a good lens. 

Although it must now be clear that this is not a very practical system 
of imaging there are some circumstances in which it has proved useful. 
For example the target of an X-ray tube cannot easily be examined for 
damage because it is permanently sealed inside a vacuum chamber and 
the ' window ' through which the X-rays emerge is often a thin metal 
roil which is opaque to visible light. X-rays cannot be focused by 
material or electromagnetic lenses and hence, if we wish to use the 
X-rays emerging to image the target, the only direct way open to us is 
to use a pinhole camera. Fig. 3.4 shows a series of pinhole photographs 
taken some years ago by Mr. W. Sutherland of the Velindre Hospital, 
Cardiff, showing progressive deterioration of a target under electron 
bombardment. It is interesting to note that here the target is being 
bombarded by electrons and is emitting X-rays so that the object is 
really * self-luminous '. The final picture is a normal photograph 
taken when the tube finally ceased to work and was taken apart. 
Fig. 3.5 (a) is a diagram of the experimental arrangement and fig. 
3.5 (b) is a photograph of a section of the ' pinhole ' used which, of 
course, has to be a massive piece of lead to provide the necessary 
opacity to the X-rays. A further point of interest is that the electrons 
are emitted from a hot spiral filament and, under the influence of the 
high voltage, they travel in straight lines; it is an emission microscope 
(next section) but with unit magnification. The spiral ' image ' of the 
filament on the target can clearly be seen in the photographs. 

3.2. Field emission microscopes 

Because of the simple geometry by which they produce images, two 
microscopes will be described. The first is the true field emission 
microscope and the second is the closely related field-ion microscope. 



DIRECT RECOMBINATION PROCESSES 



61 



In the field emission microscope the radiation used in the imaging 
process is electrons and it turns out if one examines the consequences 
of certain aspects of quantum mechanics that it is possible for a cold 
surface to emit electrons provided that a potential gradient of the 
order of 5 x 10 9 volt per metre is produced. On the face of it this is 
an enormous value but it turns out that if the specimen is small enough 
it can be achieved. The specimen is a metal wire the end of which 
has been polished (usually by electrolytic means) to a fine point which 
may be much less than one micrometre in diameter. This point is 
placed in a vacuum chamber and there is a fluorescent screen perhaps 
10 cm or so away from the point. The application of a potential 
difference of only a few kilovolts between the point and the screen can 
create the necessary potential gradient close to the point of the wire. 
In practice the wire is usually heated and, since the chamber is 
evacuated, any electron emitted is accelerated very rapidly by the field 
away from the point of the wire. It will travel in a straight line and 
of course create a glow when it hits the screen. The emission of 
radiation from the tip of the wire will depend very much on the atomic 
distribution at the tip and the result is that a magnified reproduction 
of the atomic geometry of the surface of the wire is produced on the 
screen. If the tip (fig. 3.6 (a)) is assumed to be spherical and of 
radius r, then the effective magnification, since the electrons are 
emitted normally from the surface and behave as though emitted from 
the centre of the hemisphere, is simply the ratio of the distance from the 
tip to the screen divided by the radius. If the screen distance is 
001 metre and the radius of the tip is 0-01 micrometre, a magnification 
of a million is achieved. 

Fig. 3.6 shows an optical analogue of the principle. An extremely 
small bright source of light was set up about 15 metres from a screen. 
The object, consisting of a piece of twin flex, was placed successively 
nearer to the source; the magnification achieved by this ' straight-line ' 
imaging is obvious but it is also clear that the wavelength relationship 
to the size of the object is having an effect and, in 3.6 (c) particularly, 
very strong evidence of Fresnel diffraction effects can be seen. 

It can be seen that this system very closely resembles the pinhole 
camera in its geometry and mode of operation. The other microscope 
mentioned above — the field ion microscope — operates very much on 
the same principle except that here the imaging radiation is positive 
ions. The geometry of the system is very much the same but the wire 
is charged positively and a trace of gas (usually helium) is admitted 
to the chamber. Ionization occurs and it is the helium ions which 
are accelerated away from the tip to produce the image. 



62 



IMAGES 



D ► 



(a) 




Magnification' ~ * ~ 



(b) 



(<) 




Fig. 3.6 {cont. opposite). 



DIRECT RECOMBINATION PROCESSES 



63 



(d) 




Fig. 3.6. (a) Geometry of field-emission and field-ion microscopes, (b), (c) 
and (d) Optical analogue showing how large magnification can be produced 
if an object is placed very close to a tiny source of light. The pictures also 
show Fresnel diffraction fringes round the object. 

The question might be asked why the ion microscope has any 
advantage over the field emission microscope and the answer is tied 
up with the resolution. In order to obtain a really good resolution 
it is essential that the pattern of electron or ion beams leaving the 
surface of the hemispherical end of the wire remains completely 
unchanged except for size in transit to the screen. It turns out in 
practice that random thermal motion of the atoms in the tip is the 
biggest limitation and tends to produce dizziness in the resulting image. 
In order to avoid this problem the specimen needs to be cooled to as 
low a temperature as can conveniently be obtained and of course this 
in itself makes the electron emission process somewhat difficult. The 
result is that the field ion microscope with the tip of the wire cooled 
by liquid nitrogen or some other coolant produces the best resolution 
and, in fact, is capable of revealing details down to about 2x 10 -10 m. 
This is quite sufficient to reveal the patterns of atoms, for example, in 
most metal structures. Fig. 3.7 shows a field-emission micrograph 
and a field-ion micrograph. 

3.3. Lenses for light 

In order to demonstrate the transition between the pinhole and the lens 
as image-forming systems, there is a well known experiment (fig. 3.8) 
in which several pinholes are used in the card to increase the amount 
of light transmitted while retaining the clarity of the image associated 



64 



IMAGES 



DIRECT RECOMBINATION PROCESSES 



65 



GO 




(/>) 




Fig. 3.7. (a) Field-emission micrograph from tungsten surface; the labels 
refer to crystallographic directions, (b) Field-ion micrograph from tungsten 
surface. (From Introduction to Modern Microscopy, H. N. Southworth, 
Wykeham Publications, 1975.) 




(w 



(c) 



Fig. 3.8. Geometry of demonstration of transition from pinhole to lens. 
(a) 3 pinholes, (b) 3 pin-holes with prisms to deviate beams, (c) 3 pin- 
holes with lens. 

with the small hole; the images (fig. 3.9(a)) are not then coincident 
but can be made so by placing a prism of a given angle and in the right 
orientation over each hole (fig. 3.9(b)). Investigations soon show 
that, as the number of holes is increased, each with its appropriate 
prism, the necessary prisms turn out to be effectively sections of a 
single lens. If the number of holes is increased until they merge with 
each other and the prisms are all replaced by a single lens the maximum 
use is made of all the light falling on the area covered by the lens 
while enhancing the sharpness of the image associated with the original 
pinhole (fig. 3.9 (c)). The clarity of the image of fig. 3.9 (b) must 
depend on the precision with which the prisms are made and adjusted 
and this requirement is carried through when a single lens is used: 
the optical precision of the shape of the lens surfaces is the primary 
factor governing the sharpness. One of the limitations on the sharp- 
ness of the image which were discussed for the pinhole camera still 
applies, namely the effect of the pinhole being too small. The effect 
is paralleled by the whole lens aperture being so small that its rim 
introduces noticeable diffraction effects itself. This aspect of lens 
behaviour is treated in section 5.1. 

The approach to the operation of a lens via the pinhole camera is 
acceptable as a starting point but is not adequate to enable us to 
explain all the phenomena associated with lenses. A more valuable 
way of thinking about the operation of the lens involves consideration 



66 



IMAGES 




<«) 




(b) 

Fig. 3.9 (com. opposite). 



DIRECT RECOMBINATION PROCESSES 



67 




(c) 
Fig. 3.9. (a) Photograph taken with system of fig. 3.8 (a), (h) Photograph 
taken with system of fig. 3.8 (b). (c) Photograph taken with system of 
fig. 3.8 (c). 

of the phase changes which it introduces into the patterns of scattered 
waves. 

In fig. 3.10 we have chosen one point on the object and the corre- 
sponding point on the image produced by the lens. What the lens 
does, in effect, is to ensure that by whatever path a wave travels from 
P to P' the time taken will be identical; in other words, where the air 
path is larger (e.g. PAP') the glass path is shorter and vice versa 
(e.g. PBP'). Since light travels more slowly in the glass the two effects 
can be used to compensate precisely. If the time taken is identical 
then all the parts of the wave arrive in phase at P' regardless of the 
nature of the illumination. Clearly there can be no other point on the 
screen at which all the waves from P will arrive in phase and so we build 
up a one-to-one correspondence between points on the image and 
points on the screen— just as we did for the single small pinhole. 

It may come as something of a surprise to realize that the resulting 
image is, in my terminology, the result of interference effects between 
the waves as rearranged by the lens. We have stressed that the waves 
arriving at P' will all be in phase and so will add up to reproduce the 
image of P. Why is none of the light scattered by P reaching any 



68 



IMAGES 




Fig. 3.10. A lens ensures that the time taken by the radiation to travel by 
any route such as PAP', PBP' etc. is the same for a given pair of points 
P and P'. 

other point but P'? To answer this question it is best to consider P 
to be a single ho.le in an opaque screen. A perfect lens will then 
produce what is essentially a single bright spot on the screen and no 
light anywhere else. What happens, of course, is that at P' all the 
waves are in phase and so we get an interference maximum and 
everywhere else the phase relationships are so random that the waves 
effectively cancel each other out and we have interference minima. 
If you have difficulty in understanding this last point, think about all 
the possible phases of waves arriving at any other point of the screen 
in the following way. If all possible phases exist in the collection we 
can always divide them up into pairs which differ by -n radians, and each 

pair (e.g. -H — , — h— etc.) gives zero resultant. This will not be 

true in the vicinity of the image point and this complication will be 
discussed later (section 5.1). The same argument can now be applied 
to each point on the object. 

Thus the remarkable function that the lens performs is to ensure 
that all the phases of the waves falling on it are adjusted in such a way 
that this one-to-one correspondence arises. The fact that lenses are 
so common and that they are relatively easy to make tends rather to 
blind us to the precision and elegance of their operation, 



DIRECT RECOMBINATION PROCESSES 



69 




Fig. 3.11 



Lens L 
The lens as a phase adjuster. 



We should develop the idea of the lens as a phase-adjuster one stage 
further before leaving it. It is important always to bear in mind that 
we are dealing with electromagnetic waves in three dimensions. Let 
us first consider what actually happens when a single point source is 
imaged. Fig. 3.1 1 is a sectional view in which the function of the lens 
as a phase adjuster changing the curvature of the wave fronts emitted 
by P becomes clear. It performs this function for every point being 
imaged and it is only when one considers the combined effect of the 
waves emitted from all points of the image plane to produce an 
extremely complicated resultant set of waves at A, that one begins to 
recognize the remarkable nature of the action of the lens. The 
technique of considering the wave fronts at A phase-modified by the 
distribution of delaying material (glass in this case) in the plane L is a 
very powerful one which when translated into mathematical terms 
forms the basis of modern imaging theory. (We shall say a little 
more about this in section 5.4.) 



* The behaviour of a lens as a phase-adjuster can be related to its 
behaviour as described by the conventional formula of geometrical optics 
in the following way: 

In fig. 3.12(a), consider the spherical waves diverging from the 
point P and falling on a plano-convex lens whose focal length /is equal 
to the distance from P to the lens. The lens has been drawn with a large 
thickness so that we can see what is going on inside it, but it is to be 
regarded as a thin lens. We have also drawn the angle OPA large 
but, as in geometrical optics, our result will only apply to paths making 
small angles with the axis. Let R, be the refractive index of the air, 
n., the refractive index of the glass and t the thickness of the lens on its 
axis. 

We know that, by definition, the wave fronts must become plane 
parallel surfaces normal to the optical axis to the right of the lens since P 
is at the focal point on the axis. Consequently the optical path 



70 



IMAGES 



(a) 







Jj> 


^ 


\A" 


- 


















I \ 




V 










\ 




i V 










I ° 


|0' 


\d 


0"' 










I 


I 


I 










"1 


I / 


I n 


• '/ 


"i 










/ 
I 


// 


I 
i 







(/>) c 




Fig. 3.12. (a) Geometry of lens acting as phase adjuster converting a 
spherical wave radiating from the back focal point P into a plane wave front 
0"'A'". (b) Geometrical relationships for a spherical surface. 



POO'CO'" must be identical with the optical path PAA'A"A'" if the 
phase-adjustment is to produce these parallel wave fronts. 
Along the centre line the total optical path is 

t h ?0 ; H.OO " 

= nj\ n..t 

Along the path PAA'A'A"' the total optical path is 

/»iPA-/»,AA' I n.,A'A" • /i,A"A'" 
= n l f~n 1 s ' n-if' -. -»,(/-/') 

where s is AA', /' is the lens thickness A 'A" which, since (as already 
stated) the lens is supposed to be thin and the paths at only small angles 
to the axis, may be taken as the thickness measured parallel with the axis, 
i.e. to be equal to OO". 

These total optical paths must be equal. That is 

n,l r ■ ■ rht = nj-r n } s+ ;i 2 f'— n^t-n^l' 
('h-n l )(t-t') = n l s. 






DIRECT RECOMBINATION PROCESSES 



71 



Now let us apply Pythagoras's theorem to the triangle POA' and make 
the substitution OA- h. Then 

(PA') 2 = (PO) 2 ' (OA') 2 

and hence 

(/+^=/ 2 f/r; 

P~2fs+s i =P+h\ 

Now if our restriction to paths making small angles with the axis is again 
recalled, it is clear that s will be very small compared with /and hence 
its square may be neglected and we find 

s = h i /2f. 

So the equality of optical path lengths is subject to the condition that 

(n,- rh )(t-t') = r h h*l2f. 

Now let us suppose that the lens surface is spherical — as it would be 
in the case of the thin lens studied in geometrical optics — and consider 
fig. 3.12(6). 

C is the centre of curvature of the lens surface and R its radius. We 
apply Pythagoras's theorem to the triangle CA"0" and obtain 

(CA")- = (COT '-(A'OT 
which, again subject to (he conditions already specified, is approximately 
replaced by 

/? 2 = [/?-(/ -OH/; 2 

R*=R* (t-t'f-2R{t-t') lr 

and again we can neglect (t-t'f as both / and /' are likely to be small 
compared with the other quantities and so we arrive at 

t-t' = h-\2R, 
and the condition becomes 

( ll ,-n l )lr2R = n l hV2f 
or 

(//,-/;,)//? = «//. 
You may recognize this as the formula given by the more conventional 
geometrical optics approach using the same kind of approximations. 

3.4. A hybrid technique — the zone plate 

The pinhole camera was introduced as a means of imaging by discarding 
a large part of the information that falls on the card leaving, in effect, 
just one set of data which is sufficient to produce a decipherable image 
but lacks intensity. The lens was introduced as a means of using all 
the information on the card by rearranging the phases so that all the 
light waves cooperate to produce one-to-one correspondence between 
object and image. There also exists an intermediate technique which 
involves discarding about half the information and retaining the rest 
in such a way that it cooperates to produce a relatively bright image. 
This is the so-called zone plate. It is extremely interesting as an 



72 



IMAGES 



elegant example of optical theory (and is useful as a basis for explaining 
the ideas of holography in section 4.3); only recently have important 
practical applications in image formation been developed largely 
because of manufacturing difficulties which— as we shall see later— 
the technique of holography has itself provided a means of overcoming. 
Let us first consider the problem of imaging a single point. We 
saw that the lens adjusts the optical length of all possible paths between 
the object and image points to be identical by inserting dielectric 
material of varying thickness. In order to achieve constructive 
reinforcement it is not necessary, however, for the optical path lengths 
to be identical. If the optical paths differ only by multiples of a 
wavelength of the radiation concerned the phases will differ only by 
multiples of 2n: there is no practical difference between phase differ- 
ences of 0, 2tt, 4tt, etc. (Provided that the temporal coherence is 
sufficiently high, i.e. the coherence length is great enough.) In fig. 3.13 
the paths POP' and PAP' differ by 1 wavelength, the paths PAP' and 
PBP' differ by 1 wavelength and so on. Thus waves travelling by all 
these routes arrive effectively in phase at P' just as though a lens had 
been placed in the plane OE. 




Fig. 3.13. Geometry of a zone-plate. Paths PAP' and PBP' differ in length 
by one wavelength. 

Thus, if we devise a mask which effectively cuts out those parts of 
the wavefront which would travel via paths intermediate between POP' 
and PAP' etc. we shall have achieved our aim of eliminating some of 
the information and leaving the rest in an interpretable form. As 
described so far, our screen would consist of a series of very thin 
annular apertures and very little light would be transmitted. Fresnel 
showed that, in fact, the annuli can be made very much wider, provided 
that certain conditions are obeyed, without upsetting the optical 
behaviour, and hence the amount of light transmitted can be quite large. 
The full theory is rather beyond the scope of this book but is based on 
the notion that if we make each annulus of such a width that the change 



DIRECT RECOMBINATION PROCESSES 



73 



in path length between waves travelling via the inner edge and via the 
outer edge is half a wavelength the focusing effect is the same as for 
the very narrow annuli. But we get 50 per cent of the light transmitted 
instead of a tiny fraction. The theory is known as the Fresnel ' half- 
period zone ' theory and is given in detail, for example, in Optics by 
Smith and Thomson. Fig. 3.14 is a photograph of a Fresnel half- 
period zone plate which will act as a lens and fig. 3.15 is a photograph 
taken with the camera lens replaced by a photograph of fig. 3.14 on 
35 mm film in which the overall diameter was 002 m. 

Further consideration of fig. 3.13 will show that a zone plate with 
narrow annuli will not have just one ' focal length '. There will for 




Fig. 3.14. Fresnel half-period zone plate. 




.. 



FRESNEL 



(b) 

Fig. 3.15. (a) Contact print of a line object, (b) Photograph of the line 
object using the zone plate of (a) in place of the camera lens. 



74 



IMAGES 



example be a point P" where all paths differ by 2A, a point P'" where 
they differ by 3A and so on. There will also be points — P', — P", etc. 
on the other side of the zone plate which can be regarded as ' virtual ' 
foci, since any set of waves which diverge from the zone plate as though 
from these points will have the same phase-difference relationships as 
those which come to a real focus at P', P", etc. It can be shown that 
a Fresnel half-period zone plate also has the same additional foci. 
The point about the virtual focus is often omitted in studies of the zone 
plate but it is useful to introduce it here since we shall need to use the 
idea again in talking about the principles of holography. 

Although the full half-period theory is difficult it may be worth 
applying simple geometry to the given facts of the Fresnel zone plate 
to discover how the dimensions relate to the focal lengths. 



_^, 



■> 




Fig. 3.16. Further zone-plate geometry. 



* To keep the geometry simple we will consider parallel plane wave 
fronts coming in from the left in fig. 3.16 and being ' focused ' at P', one 
of the real focal points of the zone plate OB. 

Since this is a half-period zone plate the central opaque zone of radius 
r must be such that the optical path AP' is half a wavelength greater than 
the optical path OP' (assuming that P' is the most distant of the focal 
points). 

Using Pythagoras's theorem we can see that 



(AP') 2 = r 2 r- 



or 



/ 2 -M/-A 2 /4 = r 2 ~/ 2 . 

Now A 2 is a very small quantity compared with everything else so we can 
ignore it and write 

f=rVK or r=V(/A). 



DIRECT RECOMBINATION PROCESSES 



75 



Here r is the radius of the inner edge of the first half period zone. Follow- 
ing the same kind of reasoning we could find the radius of the inner 
edge of the plh half period zone r p for which 

(/ p\i2T = r*~P 

and, using the same approximation, we arrive at 

f=r p */p\, or r p =s/(pfX). 

Finally we could extend the reasoning to other focal lengths. For 
example the next shortest focal length/' (fig. 3.13) would involve path 
differences of 3A/2 instead of A/2 and so the path-difference equation 
would become 

(r-3p\/2y = r p *-r- 
which leads to 

f' = r p 2 /3pX, or r p = v (3p/'A). 

Similarly /", the next focal length would be r p l i5pX and so on. So 
/=3/' = 5/". 

For the zone plate used for fig. 3.15 r, = 6-7x 10 -4 m and /i = 0-7 m 
(for He-Ne laser light of wavelength 6-33 x 10 -7 m). The zone plate 
can be thought of as a kind of phase-adjuster but it operates on the 
rather brutal principle of rejecting everything that is not of the right 
phase for its purposes rather than, as the lens does, by actually changing 
the phase to suit. It is not too surprising however to find that it 
obeys the normal lens formula of geometrical optics for any of its 
values of/. 

There is another type of zone plate which has recently assumed 
considerable importance in practical optical systems. By substituting 
varying thicknesses of transparent material to alter the phase of the 
light passing through instead of opaque and transparent strips, and 
by making the variation in thickness continuous, rather than like a 
step function, it proves to be possible to produce a zone plate with 
only one real and one virtual focal point. Ft also uses practically all 
the light falling on it. In other words, it behaves exactly like a lens, 
but can be extremely thin even if the aperture is very large. In systems 
in which scatter from minute specks in the glass or absorption in the 
thickness of the glass is critical, these new zone-plate lenses may prove 
to be of great value. The problem has been of how to manufacture 
them. A technique based on holography has now been developed and 
will be described in Chapter 6. 

3.5. Lenses for electrons 

One of the most powerful image-forming devices is the electron 
microscope and this conforms to the general principles outlined so far. 



76 



IMAGES 



DIRECT RECOMBINATION PROCESSES 



77 



Electrons are scattered by the object and the resulting pattern needs 
to be recombined or decoded. Electrons are very rapidly absorbed 
by most solid materials, so that the production of lenses comparable 
with those for visible light is not very practical. Fortunately electrons, 
being charged particles, are affected by both electric and magnetic 
fields. As a result electron ' lenses ' may be produced. Their actual 
operation is highly complicated but, fortunately, we find in practice 
that, if the aperture is kept very small, their behaviour can be regarded 
as phase adjusters that ensure a one-to-one correspondence between 
points on the object and image planes — though in the case of the 
electron lenses the inversion effect is replaced by a rotation of the whole 
image which varies with the object and image distances. This is a 
minor point however as far as we are concerned in this particular study. 
Fig. 3.17 shows the actual paths taken by electrons through an electro- 
magnetic lens and it is the helical nature of these paths that gives rise 
to the rotation of the image. 




Fig. 3.17. Helical paths of electrons in electron-microscope lens. 

From the many practical problems that arise in translating what 
appears to be a very simple device in principle into the very powerful 
tool that the electron microscope has become, we shall discuss only 
three. The first is that the object being studied needs to be inside a 
high-vacuum system and this immediately rules out many possible 
objects. Great ingenuity has been employed in finding solutions to 
this particular problem, for example, by preparing cast replicas of 
delicate objects that are then robust enough to be placed in the vacuum 
system; these are, however, outside our main concern and are discussed 
in detail in the Wykeham monograph No. 33 entitled Electron 
Microscopy and Analysis by P. J. Goodhew. 

The second problem is that, as with optical microscopy, the image 
lies in a plane even though the object may be three-dimensional. 






The one-to-one correspondence which we have come to recognize as 
the hallmark of a lens system relates points in two planes and hence 
only one plane of the object can be properly imaged; the others will 
be ' out of focus \ This effect — termed ' depth of focus ' — is familiar 
to photographers and indeed arises whenever a lens system is used in 
the conventional way. 

The third problem that must be mentioned here is that of lens per- 
fection. Unfortunately — largely because of limitations on the precision 
of mechanical machining operations — electromagnetic lenses of large 
aperture are extremely difficult to make. In most practical electron 
microscopes the effective aperture is very small indeed relative to that 
possible with optical microscopes. Consequently, although electrons 
have the right kind of wavelength for imaging atoms, the practical 
apertures available restrict resolution to relatively large groupings of 
atoms. 

The problems of depth of focus and of resolution limits are both 
discussed in more detail in Chapter 5 and some solutions are described. 

3.6. Imaging by scanning 

We should now remind ourselves once more of the basic problem that 
is our main concern in this chapter. In the first stage of the image- 
forming process radiation is scattered or emitted by the object and 
every point on the receiving plane contains information about every 
point on the object. Our problem is to disentangle the information. 
The pinhole camera achieved this at each point of the image by 
eliminating the information about all other points on the object and 
the consequent disadvantage is that so much of the scattered radiation 
is discarded that the resulting image is comparatively weak. The lens 
achieves a much more satisfactory result by rearranging the information 
so that a high proportion of the radiation scattered by one point 
arrives at a particular point on the image. But suppose we are dealing 
with radiation for which no lenses exist and we seek a degree of 
magnification and a level of illumination that rules out the pinhole 
system. Is there any other approach we could adopt? Suppose that 
instead of eliminating the information about every point but one we 
adopt the opposite plan. We choose a particular point on the object 
and extract from every point on the image plane the information about 
this particular point. (A lens, of course, does just this, but it does it 
for all points simultaneously; here we propose to take each point in 
turn.) We should not be throwing information away and so would 
not have the disadvantages of the pinhole camera. We should how- 
ever need to take the object point by point in order to build up the 



78 



IMAGES 



whole image and this would mean that instead of instantaneous 
imaging the process would take time. Fortunately this is very simple 
to achieve — at least in theory. In figs. 3.18 (a) and (b) we start with a 
pinhole camera and simply move the pinhole until it touches the 
object. The pinhole can then be moved around on the object and for 
each point we shall obtain a screen-full of scattered radiation. But 
now the screen will be uniformly illuminated and, at any one position, 
we shall have information relating only to P t , or to P 2 , or to P 3 etc. 
(In fig. 3.18 (b) P 3 is the point selected.) Since the screen is uniformly 
illuminated we can place a single detector, e.g. a photo-cell, anywhere 
on it (for example at C in fig. 3.18(6)). As we move the pinhole 
from point to point P,, P 2 , P 3 etc., the cell C will produce a signal 
corresponding to the information relating to each point. 

We now wish to build up an image of the original object and this 
can be done with a replica of the moving pinhole system in which the 
pinhole might be replaced by a small lamp whose brightness is con- 
trolled by the output of the cell C (also shown in fig. 3.18 (b)). Thus 
when the lamp is at position Lj it will reproduce an intensity corres- 
ponding to the light scattered by P l and so on. In this way the 
complete one-to-one correspondence can be built up in the time taken 
for the complete coverage of the object by the pinhole. 

This is one example of the process called scanning; in practice it 
may be achieved in many different ways, some of which involve moving 
a spot of illuminating radiation systematically over the object and 
others the dividing up of the scattered radiation. It has one additional 
advantage that makes it of great practical significance and that is that 
it converts two- or even three-dimensional information about the object 
into a sequence of signals which vary in time (for example in fig. 3.18 (b) 
this would be the signal from the cell C controlling the lamp L). This 
of course is essential for transmission of information over radio or 
telephone links and for many other purposes where the simultaneous 
transmission of vast amounts of data would be difficult or impossible. 
It thus lies at the heart of all systems of recording and transmitting 
images via electronic apparatus. 

One of the earliest applications was in television when the famous 
Nipkow disc (fig. 3.18 (c)) provided an effective means of transmission. 
It consisted of a spiral of holes pierced in a rotating disc that behave 
exactly as the pinhole in fig. 3.18 (/>), traversing the object line by line. 
The receiver consisted of an identical disc moving over a neon lamp 
whose brightness was controlled by the incoming signal. Thus instead 
of moving the lamp, as in fig. 3.18 (b), a large lamp was scanned by 
the holes. 



DIRECT RECOMBINATION PROCESSES 



79 





(C) 




Fig. 3.18. (a) Pin-hole camera can be changed into point-by-point scanning 
system by moving pin-hole towards object, (b) When the pinhole is in 
contact with the object, only one point of the object can be studied at once 
and radiation from it covers the whole screen. A photo-cell at C would thus 
respond to the amount of scattering from P,, P.,, P 3 etc. as the pinhole is 
scanned across the object. If the lamp is moved in synchronism with the 
pinhole to Lj, Lj, L3 etc. and its brightness is controlled by the output of C 
the lamp will produce an image of the object point by point, (c) The original 
Nipkow disc used for scanning in early television. In this case a spot of 
light was made to scan across the object: the result is the same as if the 
pinhole had scanned across in contact with the object. 



80 



IMAGES 



In practice, modern television cameras use much more sophisticated 
electronic scanning systems, but the principle of taking the information 
point by point remains identical. Similarly in radar and ultrasonic 
systems and in certain kinds of electron microscopes scanning tech- 
niques are used. Very often the object is illuminated by a narrow 
beam of radiation so that at any one moment only one point on the 
object is scattering. If you think about it you will see that the principle 
is exactly the same as that already described. We shall discuss some 
specific applications in detail in Chapter 6. Imaging by scanning is 
not quite a direct recombination process since the image has to be 
built up point by point. It has been included in this chapter because 
it finds a place in devices such as television cameras which scan with 
such rapidity that, to a human observer, the picture appears to be 
transmitted instantaneously. 

In the next chapter on indirect recombination we confine our attention 
to techniques that underline the two distinct steps in the image-forming 
process by requiring the observer to do two separate experiments in 
order to achieve first scattering and then recombination. 






4. Principles of indirect 
recombination processes 



" More ways of killing a cat than choking her with cream " 

Charles Kingsley 
Westward Ho, ch. 20 



4. 1 . The essential problem in X-ray, neutron and electron diffraction 

The chart in fig. 1.4 shows that, if we wish to study matter in atomic 
detail — for example to reveal the positions of the carbon atoms in a 
vitamin A molecule — we must use either X-rays, electrons or neutrons. 
The problem with electrons is that, so far, the only satisfactory way 
of eliminating the effects of the various aberrations of electromagnetic 
lenses is to make the aperture extremely small. The effect of aperture 
is discussed in more detail in the next chapter but we saw in Section 1.2 
that the smallest details scatter radiation through the largest angles 
and hence if we cut down the aperture of the lens — or indeed of the 
pinhole or scanning hole — the information contained in the higher- 
angle scattering will not be accepted into the recombination system 
and so the finer details of the image will be missing. For electrons, 
therefore, the present-day limits on the perfection of available lenses 
prevent us from making use of their favourable wavelengths. As far 
as neutrons and X-rays are concerned no material or electromagnetic 
lenses are physically possible. Curved mirrors have been tried for 
X-rays but it is impossible to produce a magnification greater than a 
few hundred and even that is difficult. The magnification sought in 
order to study matter down to atomic dimensions (of the order of 
10 -10 m) is about 10 8 and hence this possibility may be rejected. What 
about the pinhole camera? We saw in fig. 3.4 that a pinhole camera 
for X-rays is feasible. Suppose we wish to reveal the atoms on the 
tungsten target what must we do? First of all we need to produce a 
magnification of the order of 10 8 and simple geometry shows that, 
even if we placed the pinhole say 0-01 m from the target, the film would 



82 



IMAGES 



need to be 1000 kilometres on the other side of the pinhole — and on 
that score alone the experiment is not very practical! But apart from 
that it is also clear that the pinhole would need to be comparable in 
dimensions with the detail to be imaged and this means a pinhole of 
atomic dimensions — which again presents problems! In other words 
pinhole camera systems can be ruled out for all the radiations in this 
region of the chart of fig. 1 .4. Scanning also turns out to be impractical 
on the grounds that the spot of radiation which is scanned over the 
object must be of atomic dimensions and, at least for X-rays and 
neutrons, there are many other practical difficulties. For electrons — 
as we shall see later — scanning has certain very valuable practical 
advantages but these are not attainable anywhere near the dimensions 
which we are considering. The only direct possibility with electrons 
is the field emission idea mentioned in the last chapter but again this 
has very severe limitations. 

The essential problem that remains, therefore, is that if we wish to 
' see ' detail on the atomic scale we must use X-rays, electrons or 
neutrons as our imaging radiation but there is no practical direct way 
of performing the recombination part of the process available to us at 
the present stage of our technological development. It seems therefore 
that we may be in a worse position than that of our audience in the third 
'silly ' experiment of Section 1.1 (page 2); we can irradiate the object 
under study and observe the scattered patch on the screen but we have 
no lens to place in the projector. Fortunately the practical position 
is not quite as bad as this and the equivalent of the patch of light on 
the screen scattered by the slide is rather more informative. Though 
the process of interpretation is indirect and may be very tedious, it is 
capable of yielding a good deal of information. The specialist subject 
described as X-ray diffraction or X-ray crystallography, which has been 
studied now for over 60 years, can be summarized as finding indirect 
ways of performing the recombination part of the process of imaging 
atoms by X-rays. 

Why is it that the patterns are decipherable when the patch of light 
from the projector clearly is not? The primary reason is that the light 
used in an ordinary projector is both temporally and spatially incoherent 
whereas we saw in fig. 2.19 that if we use visible light which has some 
degree of spatial or temporal coherence, or both, the pattern ceases 
to be completely formless. Fortunately, it is possible to produce 
X-rays with reasonable degrees of both temporal and spatial coherence 
and hence scattering patterns which contain usable information may 
be produced. The second reason why a solution is possible is that the 
arrangement of atoms in almost all solid matter has at least some 



INDIRECT RECOMBINATION PROCESSES 



83 



degree of regularity, and in a very high proportion the regularity is 
very marked. Thus the scattering patterns might be expected to be 
more like those of {b) or (c) in fig. 2. 19, rather than those of (d) or (e). 
Fig. 4.1 shows two of the earliest X-ray diffraction photographs and it 
is obvious that they do continue a good deal of information. 

It might be helpful at this stage to make a brief historical digression 
(though for a fuller account of the history of X-ray diffraction the 
reader is referred to Wykeham monograph No. 13, Crystals and 
X-rays by H. Lipson). The theme of the present book is the unity of 
the principles underlying imaging and diffraction techniques but it 
must be stressed that this unified view is only possible with hindsight. 
It has only begun to emerge over the last twenty years or so of the 
sixty odd years since the pattern of fig. 4.1 {a) was produced by Fried- 
rich, Knipping and von Laue in Munich in 1912. Within a very few 
months of the discovery of the process a theory which accounted for 
the patterns had been worked out but it was in a form which did not 
readily lend itself to practical use. It fell to W. H. and W. L. Bragg 
a year or two later to develop the beautifully simple but revolutionary 
idea that led to the rapid development of the practical use of X-ray 
diffraction in revealing atomic detail. This was the concept enshrined 
in what is now called the Bragg Law. This was certainly the most 
important single break-through and led to an avalanche of new 
discoveries in metallurgy, chemistry, biochemistry, molecular biology 
and many other sciences. Yet, paradoxically, it placed such emphasis 
on the importance of regularity in the diffracting process that it may 
even have been one of the factors which delayed some of the later 
developments. In particular the realization that the interpretation 
of scattering from irregular objects is also possible and of the links 
between diffraction and microscopy seemed to take some time to 
emerge in their present powerful form. However, it was W. L. Bragg 
himself who first (in 1939) began to realize the power of regarding the 
techniques of X-ray diffraction as parallel with image-forming tech- 
niques in visual optics. In conformity with the theme of the book, 
the principles of the solution of X-ray diffraction problems will now 
be presented as a branch of physical optics, but of course this is only 
one of several approaches. It has a second advantage for our present 
purpose in that it can be presented visually through optical analogues 
and hence avoids the introduction of somewhat difficult mathematics. 

Before discussing solutions let us look in a little more detail at the 
kinds of problems that may arise. The photographs of fig. 4.1 are 
both of the type called Laue photographs. They are produced by 
irradiating a crystal with a beam of X-rays and recording the scattered 



84 



IMAGES 




V 



(a) 






C6) 

Fig. 4.1. (a) The first X-ray diffraction photograph; the crystal is copper 
sulphate. (6) An early X-ray diffraction photograph obtained by passing a 
fine beam of X-rays along an axis of symmetry of a crystal of zinc sulphide. 
From Friedrich, K nipping and Laue. Sitzungsbertchte der Koniglich Bayer- 
ischen Akademie der Wissensehaften, Munich, 1912. 



INDIRECT RECOMBINATION PROCESSES 



85 



radiation on film. The fact that the material was in the form of a 
crystal means that the object is extremely regular and von Laue and 
his colleagues thought of it as behaving like a three-dimensional 
diffraction grating. (Fig. 6.24, p. 163 is a beautiful electron micrograph 
of a virus crystal which illustrates the kind of regularity that occurs in 
crystals. In the case of zinc sulphide — the material for the photographs 
of fig. 4. 1 (b) — the repeat unit, instead of being a complete virus unit 
with tens of thousands of atoms in it, consists of only eight atoms — four 
each of zinc and sulphur — and so is much too small to be revealed by 
electron microscopy, though the principle of regular repetition in three 
dimensions is the same). 

The radiation in both photographs was ' white ' radiation — in other 
words it consisted of a wide band of wavelengths just as visible white 
light consists of a wide range of wavelengths. In our terms it was 
temporally incoherent. For 4.1 (a) it is clear that the spatial coherence 
was not very great either — as evidenced by the large spots. (Fig. 2.19 
(w)-(r) is an optical analogue which may help to clarify this point.) In 
fig. 4. 1 (/>) however the spatial coherence was greatly improved. The fact 
of temporal incoherence means that each spot has probably been 
produced by a different wavelength and, since we do not know what 
these wavelengths are, this greatly adds to the difficulty of interpretation. 
If fig. 2.19 (m) were in colour we should see that different parts of the 
pattern were in different colours and of course the colour would reveal 
information about the wavelengths contributing; in the X-ray case we 
have no such information. This kind of photograph, however, can 
be used for revealing information about the symmetry of the internal 
structure though, on the whole, it is less useful than photographs taken 
with temporally coherent (monochromatic) radiation. 

Fig. 4.2 (a) shows a photograph for zinc sulphide taken by a so-called 
precession camera (fig. 4.2 (b)). It would be out of place to discuss 
the very complicated geometry of such cameras here. The complexity 
arises simply because we are dealing with details (atomic spacings) 
which are very similar in size to that of the wavelength of the X-rays 
used (0-15 nm) and hence the angles through which radiation corres- 
ponding to these details is scattered will be very large (refer back to 
fig. 1 .3). Thus, ideally, we should record the information on a spherical 
film surrounding the crystal. The obvious practical problems of 
creating and using spherical films need no elaboration! The complex 
cameras used by crystallographers are various alternative solutions to 
the spherical film problem. The problem is comparable to the 
geographer's production of projections of the Earth's surface in an 



86 



IMAGES 




(a) 



X-roy 

COllimolor 



"two of the 

"windows' 




(b) 

Fig. 4.2. (a) ' Precession ' photograph, an X-ray photograph in which the 
geometrical complexity lies in the camera and the pattern is relatively simple. 
The specimen is a crystal of zinc sulphide, (b) A precession camera. 



INDIRECT RECOMBINATION PROCESSES 



87 



atlas (Mercator's etc.) and permit the crystallographer to record the 
information he needs in a form that can be deciphered. 

In the precession camera the crystal is made to precess about its 
axis and the flat film also undergoes a complicated motion which puts 
it in a position tangential to the ideal spherical film at the moment 
when any particular spot is being recorded. 

When all the purely routine geometrical transformations have been 
sorted out, the photograph of 4.2 (a) can be shown to correspond very 
closely with the kind of pattern that would have been produced if 

(a) a beam of radiation of wavelength 1 1000 of that used had 

(b) been directed at the crystal and 

(c) the pattern over quite a small range of forward angles had been 
recorded on a flat film (fig. 4.3). 




\ 



Fig. 4.3. Geometry for X-rays of wavelength (say) 10 -13 m (10 -3 A) and 
atoms in zinc sulphide, or for light and holes a millimetre or so apart. 

Except for the wavelength difference, the arrangement just described 
is very like those we used in producing optical diffraction patterns 
(e.g. fig. 2.15) and so we can illustrate X-ray diffraction patterns with 
an optical parallel or analogue. 

Fig. 4.4 (a) shows a pattern made with visible light of wavelength 
approximately 7 X 10" ' m from a set of holes in a piece of card 



(«) 







sx 



IMAGES 



(*) 



(C) 




~^~ 



(d) 




Fig. 4.4. (a) An optical diffraction pattern of a mask of holes representing 
the atomic positions in zinc sulphide with the central peak obliterated to 
match the X-ray photograph of fig. 4.2 (a), (b) The mask used for (a). 

(c) Model of zinc sulphide with shadow in a parallel beam of random direction. 

(d) Model of zinc sulphide with shadow in a parallel beam in the direction 
relevant to the photographs of 4.2 (a) and 4.4. (a). 

(fig. 4.4 (6)). The holes correspond to the projections of the positions 
of the zinc and sulphur atoms in fig. 4.4 (d). The wavelength has been 
scaled up about 7xl0 6 times from that specified in fig. 4.3. The 
actual atoms of zinc and sulphur are about 2-35xlO _10 m apart 
and so to make a parallel we should have to scale this up to 
2-35xl0- 10 x7xl0 6 =l-6xl0- 3 ; this is almost exactly the scale 
on which the mask of fig. 4.4 (b) was made. 






INDIRECT RECOMBINATION PROCESSES 



89 



Thus fig. 4.4 (a) should parallel fig. 4.2 (a) and indeed we can see that 
this is so. These pictures establish the feasibility of the optical 
analogue approach to the understanding of problems of interpreting 
X-ray diffraction patterns which will be elaborated in the next section. 
The optical diffraction patterns are produced on the apparatus illus- 
trated in fig. 2.20 and, as in Section 2.4, we confine our attention to 
the detail in the central disc. 

One further point may need clarification. I have referred several 
times to the * projection ' of the positions of atoms. A study of the 
mathematical relationship — which involves three-dimensional Fourier 
transforms — shows that the photograph produced by the precession 
camera is related to the projection along the axis of the crystal per- 
pendicular to the plane of the photograph. Figs. 4.4 (c) and (d) show 
a three-dimensional model of zinc sulphide placed in a parallel beam 
of light producing a projection as a shadow on the screen. For (c) 
the axis of projection is chosen randomly and in (d) the axis of projection 
is that relevant to the photographs of 4.2 (a) and 4.4(a). 

4.2. Possible solutions using analogue methods 

Fig. 4.5 shows what happens if we start from one hole (which represents 
one atom) and add further ones to it to produce a complete hexagon 
(which can be thought of as representing a benzene ring — a regular 
hexagon of six carbon atoms of side about 1-4x10 '" m). It is 
important to realize that each separate hole gives the same pattern as 
that of fig. 4.5 (b) but the interference effects (which arise from the 
phase differences between the waves coming from the dilferent holes) 
break up the pattern of the single hole into regions of smaller size. 
Suppose we now take this unit and repeat it regularly as a lattice. 
The result is as shown in fig. 4.6; the key point to notice is that the 
basic pattern is still further broken up into regions but the overall 
intensity of each region is related to the overall intensity in fig. 4.5 (/). 
This is a point of great fundamental importance since it shows us 
(i) that the position and spacing of the spots in the diffraction pattern 
of a regularly repeating object (i.e. in the X-ray diffraction pattern of a 
crystal) are determined by the way in which the basic unit repeats in 
the crystal; that is to the size and shape of the unit cell. But (ii) the 
arrangement of atoms within a unit cell (which is sometimes, but not 
always, a molecule or group of molecules of the material being studied) 
modifies the relative intensity of the spots of the pattern. Thus the 
two variables of the unit cell size plus shape and of molecular size plus 
shape may be determined separately. Fig. 4.6 also gives an indication 
of the way in which the transition from one unit to many affects the 
a 



90 



IMAGES 



(a) 



(C) 



(e) 



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Fig. 4.5 (co///. opposite). 



INDIRECT RECOMBINATION PROCESSES 



91 



CO 



(At) 




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(/) 



Fig. 4.5. (6), (ci), (f), (A), 0), (/) Diffraction patterns of various combinations 
of holes forming parts of a simple hexagonal arrangement, (a), (c), (e), (#), 
(/*), (k) The masks used. From An Atlas of Optical Transforms, by G. 
Harburn, C. A. Taylor and T. R. Welberry, by permission of G. Bell & Sons 
Ltd. 

diffraction pattern. Figs. 4.6 (e) and (f) relate to a lattice with a fairly 
large unit cell so that the spots in the diffraction pattern are fairly 
close together. It is not too difficult, with half-closed eyes, to see the 
underlying diffraction pattern of one hexagon. In figs. 4.6 (g) and (h) 
the spacing in the lattice is smaller and much more like the kind of 
relative spacing that would occur in a crystal: it is much more difficult 
to see the diffraction pattern of the hexagon, and this ties up with 
experience with X-ray diffraction and crystals. 

Four times in the last paragraph I have used the term ' unit cell ' 
which is a piece of crystallographer's jargon that perhaps needs 
explanation. Let us begin in two dimensions: fig. 4.7 (a) and (c) show 
two examples of repeating patterns that might occur in wallpaper. 
In each case we have drawn in the outline of a basic parallelogram by 
choosing identical features in the pattern. Notice that the same basic 
shape arises whatever point is chosen provided that the unit of pattern 



92 



IMAGES 



is repeated identically. We could reproduce the complete pattern by 
taking the contents of any one of the squares drawn in {a) or (c) 
and placing them down in rows and columns touching each other. 
Such a square would be called a unit cell. Notice that although 
its exact size and shape remain constant, it has no absolute position 
within the pattern — any point of the pattern will serve as the basis for 



(«) 



(0 



(e) 




(b) 



(<0 



(/) 






Fig. 4.6 {cont. opposite). 



INDIRECT RECOMBINATION PROCESSES 



93 



(*> 








(/') 



Fig. 4.6. Mask and diffraction pattern for: (a) and (h) two hexagons, 
(c) and (ct) four hexagons, (e) and (/) Large'numbers of hexagons on a square 
lattice, (g) and (//) Hexagons and a smaller lattice than in (e). 



(a) 



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Fig. 4.7. (a) Wallpaper pattern showing alternative positional choices for a 
given unit cell size and shape, (b) Diffraction pattern of (a), (c) Wallpaper 
pattern showing alternative choices of shape of unit cell, (d) Diffraction 
pattern of (<). 



94 



IMAGES 



the cell. Figs. 4.7 (b) and (d) show the diffraction patterns of the two 
wallpaper patterns and you can see immediately that the unit cell 
impresses its features on the diffraction pattern. You may notice too 
that the longer horizontal dimension of the unit cell in 4.7 (c) appears 
as a shorter horizontal dimension in 4.7 (d). This is just one aspect 
of the relationship between the real lattice and the lattice of the diffrac- 
tion pattern that leads crystallographers to describe their relationship 
in terms of ' real ' and ' reciprocal ' space. 

The extension to three dimensions is somewhat more complicated 
but follows exactly the same principles. In a three-dimensional 
repeating pattern we pick out an identical point in each repeat unit 
and, whatever point we pick, arrive at a three-dimensional lattice of 
points one parallelepiped of which is a unit cell. Notice however that 
there is not just one choice of unit cell. Even in two dimensions there 
are alternatives (fig. 4.7 (c) has one or two different units drawn in). 
Crystallographers adopt conventions in choosing unit cells but they 
are merely for convenience and to avoid confusion; there is nothing 
absolute about the choice and the problem need not worry us here. 

Having established the basic relationships, we can now begin to see 
how X-ray diffraction patterns may be interpreted. The size and 
shape of the unit cell can always be determined directly from the 
position of the spots on the X-ray photograph — assuming that the 
wavelength and parameters such as the distance from the crystal 
to the film are known. But how may the relative intensities of the 
spots be related to the shape of the molecule or other unit? At 
first sight it may seem that it should be possible to calculate one from 
the other but this does not turn out to be so. Why? Simply because 
we are unable to record the relative phases of the X-ray beams and so 
part of the information is thrown away by the process of photography. 
As we saw in Section 2.6, there is no known way of recording the 
relative phases and so direct ' focusing ' of the scattering pattern by 
merely performing a computation is not possible. 

You may recall that in Section 1.1 we pointed out that there are 
only two ways of focusing any image. One is to know all the para- 
meters of the system (focal length of lens, distance from object and 
image to lens, etc.) and it can be shown that losing the phase information 
corresponds to losing this kind of information. The second possi- 
bility is to know what the object ought to look like and it is interesting 
to find that all the relatively sophisticated methods of solving X-ray 
diffraction problems, as well as the very straightforward ones, involve 
knowing something about the object, i.e. about the atomic arrangement 
in the structure being sought, 



INDIRECT RECOMBINATION PROCESSES 



95 



We shall consider very briefly some of the various techniques to see 
how this principle works out. The very simplest set of structures to 
solve are the so-called ' no-parameter ' structures. An example might 
be metallic tungsten, which consists of a repetition of just one 
atom at the corners of a rhombohedral cell with eight of its edges 
2-73 x 10 -10 m and four 3T6 x 10 -10 m (though for their own curious 
reasons crystallographers often describe it in terms of one atom at the 
corner and one at the centre of a larger cubic unit cell (fig. 4.8) with 
sides 3T6x 10 -10 m). In this example the shape and size of the unit 
cell could be determined directly from the X-ray photograph, as we 
saw a few paragraphs ago. Then from our knowledge of the density 
of tungsten and its atomic mass, a very simple division of the atomic 
mass by the volume of the unit cell would tell us that there can only be 
one atom in each rhombohedral cell and so the structure is completely 
determined. In most cases however the content of the unit cell is 
more complex and the next simplest method is that of trial and error. 
By using knowledge of other similar compounds that have been solved 
already or by using chemical knowledge of what atoms are likely to 
be attached to each other, or by using knowledge of the sizes and 
shapes of atoms and how they can physically pack into a unit cell of a 
given size, or combinations of these, we can start to postulate a model. 
Chemical analysis and density measurements help us to deduce how 
many of each type of atom is in each cell and by using a knowledge 
of symmetry— which can often be determined from the X-ray patterns — 
we can devise a convincing model usually known as a ' trial structure \ 
This clearly comes in the category of ' knowing something about the 
object '. Then on the basis of this trial structure the relative intensities 
that it would give in the diffraction pattern can be calculated and 
compared with those actually obtained. If they agree then it seems 




Fig. 4.8. Structure of metallic tungsten showing the relationship between the 
rhombohedral cell with one atom at each corner and the cubic cell with an 
atom at the corners and one at the centre. 



96 



IMAGES 



likely that the proposal is in fact the solution; if they do not, modifi- 
cation has to be made until it does agree — hence the name ' trial and 
error '. 

How is the calculation actually done? Cast your mind back to the 
discussion in Chapter 2 centred round about fig. 2.10 in which we saw 
how a diffraction pattern could be built up from the fringes produced 
by pairs of points. The computer can be used to do just that: in effect 
each pair of points on the object produces a set of three-dimensional 
fringes which the computer samples at specific points and adds up the 
contributions of all the fringes. If three-dimensional sinusoidal 
fringes make your mind boggle, try to imagine one of those liquorice 
all-sorts which consists of alternate layers of black and white. Now 
imagine many more layers of much greater area. Finally imagine that 
instead of sharp black and white alternations the layers blend gradually 
into each other so that a plot along a line perpendicular to the planes 
would show a variation of intensity which was sinusoidal instead of 
square wave (like a set of battlements) as it would be for the original 
liquorice all-sort! Now you have imagined three-dimensional sinu- 
soidal fringes. 

A very powerful method that has been used to solve many of the 
more complex structures of biomolecular importance is known as the 
' heavy atom ' method. The technique is to prepare two forms of the 
crystal, one with the substance alone and a second in which each of the 
molecules has a ' heavy ' atom attached to it. * Heavy ' in this sense 
means that it scatters X-rays very strongly — and since it is the electron 
clouds that do the scattering, this means one with a large number of 
electrons — i.e. a high atomic number. For example, a molecule that 
consisted of about 150 atoms of carbon, nitrogen or oxygen joined 
together (hydrogen atoms scatter so little that they can be ignored to 
begin with) each of which has six, seven or eight electrons might have 
a single mercury atom (with 80 electrons) attached to it at some point. 
X-ray patterns of the two compounds would then be compared, 
noting which of the spots gets brighter and which less bright when 
the heavy atom is present (these spots are called 'reflections' by 
X-ray crystallographers because of the Bragg Law notion of reflection 
from planes of atoms, but in our scheme of things this could be a 
confusing term). 

This enables the relative phases of the spots to be determined under 
certain conditions and hence the structure can be ' focused ' by 
computer. 

A very remarkable and elegant aspect of our subject is bound up in 
that last 'throw-away' line; "...and hence the structure can be 



INDIRECT RECOMBINATION PROCESSES 



97 



' focused ' by computer.'* In Section 2.4 we said that the mathematical 
process needed to predict the diffraction pattern of a given object is 
known as Fourier transformation. Indeed the operation performed 
by the computer that was described only two paragraphs ago is that 
of Fourier transformation. The remarkable fact however is that 
exactly the same process is involved in predicting the object from its 
diffraction pattern. Mathematically speaking (and ignoring one or 
two provisos that ought to be made if we wished to be completely 
rigorous) the Fourier transform of the Fourier transform of an object 
is the object again. 

Thus when we have the X-ray diffraction pattern we can divide it 
up into pairs of spots and each pair can be thought of as producing 
sinusoidal fringes which when added together will give us back an 
image of the object. The catch, of course, is that we must know the 
relative phases of the spots before we can perform this operation. 

To indicate the way in which the heavy atom helps us to solve the 
phase problem consider fig. 4.9 in which all the pictures are optical 
analogues of X-ray diffraction patterns. Fig. 4.9 (h) shows the pattern 
for a single molecule of a compound called phthalocyanine and 
4.9 (d) shows the pattern for a complete crystal. Fig. 4.9 (/) shows the 
pattern for one molecule with a ' heavy ' atom (in this case rhodium) 
added and fig. 4.9 (h) shows the pattern for a complete crystal with the 
heavy atom. Only the centre regions are shown in (d) and (h) so that 
the changes in the intensity may clearly be seen. 

How does this technique tie up with the idea that we must know 
something about the object? It corresponds in fact to the trick of 
focusing on the known feature— the chip in figs. 1.2 (c) (/); in this 
case the heavy atom which is known to be present in each molecule is 
the chip and we focus on it and assume that the rest is then in focus. 

The process of focusing by computer referred to above is usually 
called ' Fourier synthesis '—though it is probably more correct to call 
it Fourier transformation— and leads to electron density contour maps 
which are really the closest we can get to seeing the atoms in a crystal. 
Fig. 4.10 shows typical electron density maps that have been arrived at 
using the heavy atom technique. 

Nowadays one of the most powerful methods of obtaining a solution 
is by so-called ' direct ' methods. These give the illusion that one is 
merely taking the data and focusing an image by computation. Indeed 
there are now computerized systems which will accept the data and 
automatically calculate and even draw the * image '. What has 
happened to the principle that one must know something about the 
object? How has the phase problem been solved? 



98 



IMAGES 



Investigation shows that the mathematical relationships that are 
used to ' solve ' the phase problem are in fact based upon some very 
important facts about the structure which are implicit in the mathe- 
matical presentation. These are, for example, that the atoms are 
likely to be spherical, that the electron density is always a positive 
and real quantity — which in physical terms means that the electrons 
are behaving effectively as purely mechanical or geometrical scatterers 
and do not change the phase in any physical way. The remarkable 



(a) 





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(c) 



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Fig. 4.9 (conf. opposite). 



INDIRECT RECOMBINATION PROCESSES 



99 



(tf) 



. 



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(A) 



Fig. 4.9. Optical analogues illustrating the possible solution by the ' heavy 
atom ' method of the structure of phthalocyanine. (a), (b) Mask and diffrac- 
tion pattern of one molecule with no metal atom, (c), (d) Mask and 
diffraction pattern of projection of crystal with no metal atom, (e), (/) Mask 
and diffraction pattern of one molecule with metal atom at centre, (g), (h) 
Mask and diffraction pattern of projection of crystal with metal atoms at all 
molecular centres. In (d) and (h) only the centre region is shown. 




Fig. 4.10. Electron density map for nickel phthalocyanine. (Robertson 
and Woodward. /. Chem, Soc, 219, 1937). 



100 



IMAGES 



thing is that these facts turn out to be powerful enough to give answers 
to a remarkably wide range of crystal-structure problems. 

There are many other highly sophisticated techniques used in X-ray 
crystallography but perhaps those that we have selected will illustrate 
the principles sufficiently well for our purpose. 

Is there really no way in which something looking more like an image 
of the atoms in a structure can be prepared? Yes there is — but it is 
only of use as an elegant curiosity since the trick can only be performed 
after the full details of the structure have all been determined. It does 
however provide a powerful confirmation of the general thesis of this 
book; that is, that all imaging processes can be divided into the two 
operations of scattering and recombination and for that reason an 
example is included here. 

The technique was originally suggested by Sir Lawrence Bragg over 
thirty years ago and was perfected some ten years later in Manchester. 
The principle is simple; we use X-rays to perform the first stage of the 
imaging operation — the scattering stage — because of their small 
wavelength. Ideally we wish to convert the X-ray beams into beams 
of visible light of the same relative intensities and phases. If this 
could be done we could then use an ordinary lens to focus the image. 
The problem lies of course in the conversion of the X-ray beams into 
light beams. 

Once the intensities of the X-ray spots have been determined, the 
structure solved, and the relative phases calculated from the known 
structure, a plate containing holes to represent each X-ray spot is 
prepared. Each hole carries a piece of mica and the imaging process 
is carried out in polarized light. If a piece of mica of suitable thickness 
is rotated in its own plane between crossed polaroids or nicol prisms 
the light transmitted rises and falls in intensity and passes through a 
zero of intensity every 90 \ (See any text book on physical optics or 
crystal optics.) By rotating the mica in each hole, the intensity can 
thus be varied to match the relative X-ray intensity associated with 
the spot; it turns out that if the mica is rotated clockwise from a 
position of zero intensity the phase of the light passing through is 180 
different from that if the mica is rotated anti-clockwise. Fortunately, 
if the structure is centro-symmetrical (that is for any point with 
coordinates .v, y, - with respect to some origin there is always a point 
— x, - y, -z) the only phases that occur are and 180 and hence 
this technique provides all that is necessary. In fact further refinements 
of the technique led to the possibility of producing any required phase 
but it would be too long a diversion from our theme to discuss the 
details here. (Sec for example Lipson, Optical Transforms.) Fig. 4.1 1 



INDIRECT RECOMBINATION PROCESSES 



101 



is an example of an image produced in this way and shows a simple 
molecule and parts of its neighbours in a crystal of hexamethylbenzene 
in which the inner hexagon is the benzene ring of six carbon atoms and 
the outer hexagon is the six carbons of the methyl groups. The 
hexagons do not appear to be regular because we are viewing them 
obliquely. One could legitimately describe this photograph as a 
photo-micrograph with a magnification of about 10 8 ; but we must 
remind ourselves that it cannot be obtained directly and is merely a 
way of presenting the result when the structure has been solved by one 
or other of the techniques of X-ray crystallography. 

4.3. Holography 

In Section 1.1 the term ' hologram * was used to describe the patch 
of light on the screen formed by a projector without its lens. We said 




Fig. 4.11. Image produced by the two-stage process of scattering X-rays 
and then recombining light beams which can only be done when the structure 
has already been determined. The structure here is hexamethylbenzene. 



102 



IMAGES 



there that the reason is simply that each point on the screen is receiving 
information about every point on the object. The word is derived 
from the Greek hobs meaning ' a whole ' and of course reflects the 
notion that the whole of the information about the object is contained 
in it. This particular kind of hologram is not, however, very useful 
mainly because of the incoherence of the light. We saw in fig. 2.19 
that the scattering patterns of objects become more detailed and 
contain more information when the degree of coherence of the radiation 
increases and it seems feasible therefore to expect that it would be 
easier to extract information from a hologram if it were made with 
coherent light. Various partially successful attempts were made before 
1960 to recombine the information contained in holograms without 
using a lens but the real breakthrough came with the development of 
the gas-phase laser; in one step there was an almost unbelievable 
increase both in temporal and spatial coherence and incidentally in 

total light intensity. 

We shall first of all examine the simplest form of experimental 
arrangement used for producing holograms and for reconstructing 
images from them and then consider two simple alternative approaches 
to understanding how the technique works. Fig. 4.12 (a) shows the 
experimental arrangement for producing the hologram. 

The object O is illuminated by a temporally and spatially coherent 
beam from the laser L x . The primary beam from a gas phase laser is 
usually only a millimetre or two in diameter and so it is normal to 
expand the beam by means of a lens system. It is essential that the 
lenses used should be of fairly high quality so that there is no distur- 
bance of the spatial coherence. Light is scattered from the object 
and falls directly on to a photographic plate P. A portion of the 
original beam from L, is directed by the mirror M so that it produces 
a uniform patch of light over the whole of the plate P. Without 
going into detail at the moment, it is clear that a complex interference 
pattern will be produced on P as a result of the superposition of all the 
various scattered waves and of the so-called reference beam from M. 

In order to reconstruct the image the developed photographic plate 
is illuminated by a beam from a laser, L 2 in fig. 4.12 (b), in such a way 
that the beam falls on the plate in the same direction as that of the 
reference beam from the mirror in the first stage. An observer looking 
into the plate from E will then see an image of the object at I and the 
striking and important point is that it is a three-dimensional image. 
In other words if the eye is moved from side to side the same parallax 
effects occur as would with a real object. The effect is particularly 
pronounced if the object consists of two or more separate items some 



INDIRECT RECOMBINATION PROCESSES 



103 





(a) 

Fig. 4.12. (a) Schematic arrangement for producing a hologram. (6) Sche- 
matic arrangement for reconstructing an image from a hologram. 

of which are nearer to P than others. Fig. 4.13 shows two views of 
such a reconstructed image from different directions, and the parallax 
effect is most pronounced. 

How can the process be explained? Let us first consider a rather 
elementary approach. In fig. 4.12 (a) we have a set of waves scattered 
from O which have very precise phase relationships with each other by 
the time they reach the plane of the photographic plate at P. The 
reference beam is simply a uniform wave front also arriving at P. 
The two sets of waves will be superimposed on each other and an inter- 
ference pattern will be produced which may be recorded on the plate. 
In the reconstruction stage, 4.12(6), this interference pattern recorded 
on the plate is introduced into the replica of the reference beam 
produced by L 2 . The effect as far as the eye E is concerned is that the 
light from L 2 has been modified to resemble precisely the superposed 
interfering waves at P in fig. 4.12 (a). You will recall that we met the 
idea that all optical images are really interference patterns and the eye 
and brain accept these waves to form an image in exactly the same way 
that they would accept the interfering waves from a real object placed 
at I. The effect to the observer is therefore precisely as though a full 
three-dimensional object were actually at I. 

This explanation may or may not convince you: it is reasonable as 
far as it goes but it leaves a great many questions unanswered and of 






104 



IMAGES 




Fig 4 13 Two views of a reconstructed image from a commercially 
produced hologram illustrating the three-dimensional character of the image. 



INDIRECT RECOMBINATION PROCESSES 



105 



course a really full explanation demands quite sophisticated mathe- 
matics. The following treatment takes us a little further, however, 
and may be found useful. 

* Let us suppose that the object O is replaced by a single scattering 
point. The waves scattered from it (remember that it is illuminated by 
temporally and spatially coherent light) will simply be a regular train of 
spherical wave fronts diverging from the point. The intersection of 
these spheres with the plane of the photographic plate will be a set of 
concentric circles. Suppose that we ' freeze ' the whole system in time 
and that we consider only the spherical surfaces at intervals of 1 wave- 
length where the phase has some particular value. Then the intersection 
with the plane of the plate will be a set of concentric circles which get 
closer together as we move away from the centre (see fig. 4.14). Thus if 
we now superpose the uniform reference beam the resulting interference 
pattern on the plate will look very like a zone plate (Section 3.3) except 
that the zones will not have sharply defined edges. Now, on transferring 
to the reconstruction stage (fig. 4.12 (/>)) we have a zone plate in a parallel 
beam of light and hence a point image will be produced (a real one to the 
right of the plate and a virtual one to the left). The same argument may 
be followed through for every point on O and we can then see that the 
hologram is really an immensely complex superposition of zone plates 
each of which will produce an image of its own corresponding point on 
the object. 

Two questions immediately jump to mind. What about the multiple 
images produced by a zone plate? In the last chapter we said that a new 
kind of zone-plate lens could now be produced by holographic methods 
which had only one focus on either side. Indeed the explanation we have 
just given is at the heart of the new technique. It is the substitution of a 
sinusoidal variation for the square wave variation in blackness of the 
rings that, in fact, removes the higher orders. The effect is very much the 




Fig. 4.14. Diagram showing interaction of spherical waves with plane of 
photographic plate. 
H 



106 



IMAGES 



same as the fact that a diffraction grating in which the variation of density 
along a line perpendicular to the grating lines is sinusoidal, instead of a 
square wave, gives only one order of diffraction on either side of the centre 
instead of many. 

This last point is an important one and can most easily be approached 
by thinking of the reversibility of light waves — or the two-way application 
of the mathematics of Fourier transformation. 

Consider first of all the diffraction pattern of a double slit: it is simply 
a cosinusoidal variation in amplitude. (We say cosinusoidal here because 
there is a maximum at the centre as for a cosine curve). Fig. 4.15 (a) 
illustrates this. Now suppose we add a third slit on the axis, midway 
between the other two and suppose, although still very narrow, it transmits 
twice the amplitude of the other two. Think back to our consideration 
of the geometry of the double-slit on page 33 and you will realize that 
this third slit is on the axis and therefore contributes in phase with the 
beam from point O in fig. 2.13 everywhere on the screen. The resultant 
pattern is thus as shown in fig. 4.15 (b) — the cosinusoidal curve is lifted 
so that it is everywhere positive and the former negative regions just 
become zero. 

Now suppose we make a grating whose transparency distribution is 
like that of fig. 4.15 (/>); the principle of reversibility of light waves or the 
Fourier transform relationship tells us that its diffraction pattern will be 
the equivalent of the three slits. That is, it will be a central order with 
one order on either side each of half the amplitude of the centre one. 

Now to return to the zone plate view of the hologram. What about 
the real image? It turns out in fact that there is a real image from a 



Amplitude 




position on screen 



Amplitude 



(b) 



(c) 



CO 




position on screen 



I I 



I I I 



Fig. 4.15. (a) Cosine amplitude distribution from double slit, (6) Cosine 
amplitude distribution with constant term added resulting from triple slit, 
(c) Double slit producing (a), (d) Triple slit producing (b). 



INDIRECT RECOMBINATION PROCESSES 



107 



hologram and this can be picked up and photographed just as easily as 
the virtual one. In fig. 4.12(6) it would be at I' and could be photo- 
graphed by a camera placed at E'. It is not quite so easy to observe the 
real image because the waves are diverging from it and a camera of limited 
aperture would have to be very carefully aligned in order to see it and the 
field of view tends to be more limited. 

Holography has some very interesting applications in many different 
areas of physics and a section of the last chapter is devoted to two of 
them (6.13). 



5. Perturbations of the image 



" Yes, 1 have a pair of eyes," replied Sam, " and that's just it. 
If they was a pair o' patent double million magnifyin' gas 
microscopes of hextra power, p'raps I might be able to see 
through a flight o' stairs and a deal door; but bein' only eyes, 
you see, my wision's limited." 

Charles Dickens 

The Pickwick Papers, ch. 34 



5.1. Aperture and wavelength 

We have already hinted once or twice at the possible effects if the 
object is so small compared with the wavelength of the radiation that 
its diffraction pattern involves large angles which take much of the 
radiation outside the aperture of the instrument; but we must now 
look in a little more detail at the consequences. First let us consider 
a simple circular aperture and the problem of imaging an object which 
consists of a fairly regular arrangement of holes of different sizes. 
We shall use the instrument described in Section 2.5 (fig. 2.20 (a)) with 
lens L 3 in position so that an image of the object placed at P is produced 
at R. If we start with the instrument operating at full aperture, most 
of the diffracted waves will enter the system and contribute to the image 
and hence the image is almost identical with the object (fig. 5.1 (a)). 
Its diffraction pattern (recorded at Q) is shown in fig. 5.1 (b). Now 
if we introduce an aperture at Q which cuts out some of the diffracted 
beams and only allows the portion shown in fig. 5.1 (c) to enter the 
system, the image at R becomes like fig. 5.1 (d). Notice immediately 

Opposite, above: 

Fig. 5.1. (a) Object (mask) of holes in opaque card. (b) Optical diffraction 
pattern or transform of (a), (c) Portion of (b) allowed to go forward for 
recombination, (d) Recombined image from (c). 

Opposite, below: 

Fig. 5.2. (a) Smaller portion of fig. 5.1 (b) selected for recombination. 
(b) Recombined image from (a), (c) Still smaller portion of fig. 5.1 {b) 
selected for recombination, (d) Recombined image from (c). Figs. 5.1 and 
5.2 from An Atlas of Optical Transforms, by G. Harburn, C. A. Taylor and 
T. R. Welberry, by permission of G. Bell & Sons Ltd. 



PERTURBATIONS OF THE IMAGE 



109 



(a) 



(c) 




(b) 



(d) 



Fig. 5.1. 



(a) 



# 


'■ 

i 
i 


il 

nil !jg_ 


■ 

i 

• 


i 

•i 




m-mt 







(b) 



(c) 



9 *l 

il • 



id) 



Fig. 5.2. 



110 



IMAGES 



that it is the finest detail that is affected ; some of the very closely spaced 
holes have become blurred into a line and in the case of the shapes at 
the corners of the object the sharp angles have become rounded. 
This process continues as the aperture is still further reduced (fig. 5.2). 
In practice these limitations occur in all systems whether using 
visible radiation or not. This is one of the reasons why astronomical 
telescopes, both visual and radio, are made as large as possible— the 
other main one being that of course a large aperture gathers more 
light waves and hence is likely to produce an image that is easier to see. 
Since the stars we observe are so far away, each one bathes the whole 
Earth with its radiated waves and there is no theoretical limit to the 
improvement obtained by increasing telescope size— a telescope with 
an aperture equal to the Earth's diameter could in theory pick up all 
the radiation falling on the Earth and hence give the best image! 
Obviously there are other practical considerations that make limitations 
long before this size is reached. In the microscope, however, the 
situation is different. The objective lens of the microscope is so close 
to the object that it is possible to design a lens which will take in almost 
all the cone of scattered radiation from the object and hence the 
theoretical limit of resolution is much more closely approached with 
microscopes than with telescopes. One of the reasons why the so- 
called oil-immersion objectives (in which a drop of oil fills the space 
between the objective and the object) are used is that the oil minimizes 
the effect of the boundaries between the cover slide and the lens and 
so increases the cone angle of light actually entering the system, 
thereby increasing the resolution. 

I have talked in terms of changing the size of the aperture— but a 
moment's thought and a further study of fig. 1.3 shows that it is the 
size of the aperture relative to the wavelength that matters and that 
doubling the wavelength for the same aperture is the same as halving 
the aperture with the same wavelength. For visible light we only have 
a wavelength range of between about 4x 10" 10 m and 7-5 x 10" 10 m so 
we can barely double the wavelength even if we go from deep purple 
to deep red. In other regions of the electromagnetic spectrum, 
however, big changes of wavelength can be made. If we are designing 
a radar set, therefore, a smaller wavelength will not only suit us better 
from the point of view of allowing us to encode more detailed informa- 
tion about the object, but will also allow us to use smaller reflectors. 
In radio telescopes, however, the choice of wavelength is made by the 
stellar radiating sources and we have to design our dishes of a suitable 
size to give the required resolution with the given wavelength (which 
might be of the order of 20 m). 



PERTURBATIONS OF THE IMAGE 



I I 



It will help us not only now but also in later work if we give some 
attention to the more detailed aspects of the behaviour of apertures. 
Consider, for example, a telescope used by an observer to look at a 
single star. The star is so far away that the spherical waves radiated 
by it will have a radius of curvature that is quite a reasonable approxi- 
mation to infinity! The telescope is thus effectively being irradiated by 
a parallel beam of light. The lens, of course, has the effect of bringing 
this parallel beam to a point focus in its back focal plane (assuming that 
the lens is perfect) and so we have the ideal conditions for observing 
Fraunhofer diffraction (as specified in Section 2.5). Thus if we placed 
a small circular aperture over the lens, we should see the aperture's 
Fraunhofer diffraction pattern when we look through the telescope. 
Now suppose we enlarge the aperture. The diffraction pattern of a 
circular hole is the famous Airy disc (fig. 5.3 (a)) and as the hole gets 
larger the pattern stays the same in form but is reduced in scale. 
(Some authors retain the name 'Airy Disc ' for the central peak 
only: I find it more convenient to use it to refer to the whole pattern 
of central disc and concentric rings.) The radius of the central disc 
is, in fact, inversely proportioned to the radius of the hole (fig. 5.3 (b)). 
What happens when the aperture becomes sufficiently large to coincide 
with the rim of the lens? Clearly we still have an Airy disc, albeit a 
small one, when we look at the star through the telescope. Indeed 
this is the smallest disc we can obtain with the given aperture of the 
telescope. In other words we can never see a true image of the star 
but will always see the Airy disc of the aperture. This is the origin 
of the comment in Section 3.3 that, when considering an image as an 
interference pattern, the notion that the phases other than at image 




(a) (b) 

Fig. 5.3. Diffraction patterns (Airy disc patterns) of (a) 2 mm diameter 
circular hole, (b) 4 mm diameter circular hole. 



112 



IMAGES 



points are such that they result in zero amplitude is not true at points 
close to the image point. 

Now suppose there were several stars in the field of view. Precisely 
the same argument would apply to each and we should have an Airy 
disc at each of the points corresponding to a star. They would all be 
the same size though their relative brightness would depend on the 
brightness of the stars. (Fig. 5.4 (a) is a picture of a star field and the 
Airy discs can be seen very clearly.) 







Fig. 5.4. (a) Simulated star-field with Airy pattern at each point as it would 
be imaged in a telescope of small aperture, (b) The same star-field imaged 
with a rectangular aperture over the telescope aperture. 

The process of replacing each point image of a star by an Airy disc 
of identical size and form is an example of a very important concept 
in Fourier transform theory called ' convolution \ Convolution is a 
mathematical operation which for our purposes can be thought of as a 
process of dealing a particular item to a series of positions. An 
example from everyday life might be the regular arrangement of eggs 
in a moulded tray ready to be packed in a crate. We could describe 
the arrangement of the eggs as the convolution of one egg with a set of 
mathematical points arranged on a square lattice. We could describe 
the star field as shown in fig. 5.4(a) as the convolution of two items; 
one is a picture of the star field with mathematical point images and 
the other is a single Airy disc. The fascinating point that emerges is 
that this phenomenon of convolution always occurs in diffraction 
patterns when two objects are ' multiplied \ That last clause may 
take a bit of explaining. An example that may be familiar is that of 
4 crossed ' diffraction gratings. A diffraction grating placed in a laser 
beam gives a single row of regularly spaced orders in a direction 



PERTURBATIONS OF THE IMAGE 



113 






perpendicular to the lines. A second identical grating placed on its 
own in the beam with its lines perpendicular to those of the first will 
also give a regular row of orders, this time in a direction at right angles 
to those of the first. If now both gratings are placed in the beam so 
that the beam goes first through one and then through the other the 
'crossed' gratings are effectively multiplied; the diffraction pattern 
now consists of a set of orders at the points of a square lattice — which 
can be described as the convolution of the set of orders from one 
grating with the set of orders from the other (see fig. 5.5). That this 
operation is multiplication and not addition is also illustrated in fig. 5.5. 
When the two gratings are placed side by side the result is the single 
addition of the two separate transforms translated, as always, to a 
common centre. The fact that the addition depends on phase as well 
as amplitude is demonstrated by the fringes in the central region where 
the two patterns overlap; these fringes are characteristic of the lateral 
separation of the two gratings. 

In the case of the telescope, the lens aperture is bathed in the 
scattering pattern of the star field. All the parts of this scattering 
pattern outside the aperture are lost completely and this can be 
regarded as 'multiplication by nought'. Thus if we think of the 
aperture as a function of magnitude one within the circle and of 
magnitude nought outside it, you should be able to see that we have 
indeed multiplied the scattering pattern of the star field by the aperture. 

To understand the idea of convolution completely takes a long time 
but I hope the basic idea will begin to make sense soon, as it can 
simplify enormously our approach to some of the later problems. 

We have already seen quite a few examples in earlier parts of the 
book. In Section 4.2 we talked about the separation of the variables 
of lattice dimensions and unit-cell contents. In our new terminology 
the crystal can be thought of as the convolution of a lattice of mathe- 
matical points with the contents of one unit cell. Now we have 
already met the reciprocal nature of Fourier transforms and it should 
not be too surprising now to be told that this time the diffraction 
patterns are multiplied. Fig. 4.6 in fact shows this very clearly; 
fig. 4.6 (/) can be seen to be the product of two items, one a perfect 
lattice with every point of the same intensity (which is the diffraction 
pattern of the lattice of mathematical points), and the other, the 
diffraction pattern of the unit cell contents (fig. 4.5 (/)). 

Now we should be in a position to answer the question, " What 
would happen if we were to make a telescope with a rectangular 
aperture instead of a circular one?" Clearly we should now be 
multiplying by a rectangular function and so our star field would be 



114 



IMAGES 




Fig. 5.5. (a) A diffraction grating of rectangular slits, (b) Optical transform 
of (a), (c) Cross grating produced by ' multiplying ' the grating of (a) by a 
second identical grating turned through 90 c and placed on top of (a): light 
is only transmitted where both are clear, (d) Optical transform of (c): it is 
the convolution of (b) with itself turned through 90°. (e) The grating of (a) 
' added ' to a second identical one turned through 90° by placing them side 
by side. (/) The optical transforms of (<?): the transforms also are added. 



PERTURBATIONS OF THE IMAGE 



115 



the convolution of the point-field with the diffraction pattern of the 
rectangle. Fig. 5.4 (b) shows the star field in the telescope with the 
rectangular aperture. The diffraction pattern of a rectangle now 
appears at each star point. 

One final remark needs to be made. Telescopes are, of course, 
used to observe many objects other than star fields, but the convolution 
principle remains valid. Every point of an extended object in fact 
becomes an Airy disc if the aperture is circular and the result may 
be quite complicated. It is considered in more detail later in the 
next section. 



5.2. False detail and possible misrepresentation near the resolution limit 
Almost since the microscope was invented there have been controversies 
about the fineness of detail that could be observed. Usually the 
physicists have been on one side of the fence, claiming — along the 
lines that we have already discussed — that it is impossible to image 
detail whose significant dimensions are much less than the wavelength 
of the radiation used ; on the other side of the fence have been the 
practical microscopists who have said " But look down this microscope 
— I can actually see detail that is smaller than the wavelength '*. 
What is the solution to the paradox? As so often is the case both 
claims are true. The microscopist can see detail that is smaller than 
the wavelength but it may very well be totally false and may not 
correspond to anything that exists on the object. Thus in the end the 
physicist is also right if he changes his claim to indicate that it is 
impossible to make a reliable image of detail whose significant dimen- 
sions are much less than the wavelength of radiation used. 

How can this finer detail arise? Fig. 5.6 (a) shows a pattern of four 
points arranged on a square. Fig. 5.6 (b) shows this pattern as it 
would be imaged by a system with a limited circular aperture. We 
obtain this pattern by using the convolution process described in the 
last section and placing an Airy disc pattern at each of the four points. 
Now suppose that the four points move closer together (fig. 5.6(c)): 
the Airy discs will also move closer together and a position can be 
found when the first ring of each of the four Airy patterns all intersect 
at the centre of the square (fig. 5.6 (//)). The result will be a bright 
spot at the centre and we shall ' see ' a spot that is not there. Because 
the rings of the Airy pattern have a thickness which is less than half 
the diameter of the central disc, this additional spot will be smaller 
than the discs and may well appear smaller than the normal limit of 
resolution, 



116 



IMAGES 







(:) 



• 




Id] 



Fig. 5.6. (a) Four points arranged on square, (b) Airy disc patterns at 
each of the four points: the lines represent zeros and the central discs are 
shown black, (c) Four points much closer together. (//) Airy disc patterns 
of the same size as those in (b) placed at the points of (c). The central diamond 
shaped patch shown black is the superposition of the first bright rings from 
all four patterns: note how much smaller it is than the central discs in (b). 

Fig. 5.7 shows some examples of the kinds of spurious effects that 
can occur. Fig. 5.7 (a) is the object and (b) is its diffraction pattern. 
When an attempt is made to form an image with a small aperture (c) 
which allows only a limited amount of the diffraction pattern to enter 
the system the result is (d). The condition referred to in the last 
paragraph and illustrated in fig. 5.6 has been deliberately created for 
the 5 x 5 array of holes at the centre; the additional ' holes ' which are 
much smaller than the Airy discs can clearly be seen. In (e) the 
aperture is still further restricted and the result at (/) is spurious in yet 
a different way. We now have a 3 x 3 array at the centre in place of 
the real 5x5 array ! 

Fig. 5.7 alone is an awful warning to microscopists not to believe 
all that they see, especially near the limit of resolution for their par- 
ticular system. That, in turn, means that microscopists — or indeed 
any users of image-forming systems — should always know what the 
resolution limit of their system is. 

A very beautiful practical illustration of the effects was produced 
some years ago by Mr. A. W. Agar — it is shown in fig. 5.8. On the 



PERTURBATIONS OF THE IMAGE 



17 



(a). 






Fig. 5.7. (a) Regular object, (b) Optical transform of (a), (c) Restricted 
pattern of (b). (d) recombined (c): note particularly the 'extra' holes 
which have appeared in the central square as illustrated by fig. 5.6. (e) Further 
restrictions of (b). (/) recombined (e): note that the centre square now 
appears to be 3x3 instead of 5x5. From Optical Transforms, by C. A. 
Taylor and H. Lipson, by permission of G. Bell & Sons Ltd. 



IMAGES 




PERTURBATIONS OF THE IMAGE 



19 



Fig 5 8 (a), (b) Optical micrographs of two different diatoms showing 
detail close to the resolution limit, (c), (rf) Electron micrographs of the 
identical specimens at the same magnification in the electron microscope. 
By permission of Mr. A. W. Agar, Bishop's Stortford, Herts. 

left are two optical micrographs of some diatoms in which the fine 
detail is beyond the resolution limit. On the right, by extremely 
elegant manipulation, electron micrographs— for which the detail is 
well within the resolution limit— have been made of the identical areas 
of the diatoms so that the true detail can be clearly seen. 

The effect does not only hold for regular objects, and fig. 5.9 shows 
the effect of aperture reduction on the image of Mickey Mouse! 
Again it is the fine detail and sharp edges which suffer. In modern 
optical jargon it is common to say that the ' higher spatial frequencies ' 
are excluded. This is of course a mixture of ideas and relates to the 
model put forward in Section 2.3 about the image being built up from 
sets of fringes; the object with the highest spatial frequencies— that 




(O (d) 

Fig. 5.9. (a) Outline of somewhat irregular object, (b) Optical transform 
of (a), (c) Restricted portion of (b). (d) Recombination from (c). From 
An Atlas of Optical Transforms, by G. Harburn, C. A. Taylor and T. R. 
Welberry, by permission of G. Bell & Sons Ltd. 

is with details very close together— produces fringes with the widest 
spacings and these are likely to be the first to go when the aperture 
is reduced. 

We can still use the notion of convolution, however, and in fig. 5.10 
we have deliberately chosen objects which, though continuous rather 
than being made up of isolated points, nevertheless display in recog- 
nizable form the features of the Airy discs with which each element of 
the object is replaced. In fig. 5.8 the optical micrograph of the diatoms 
show fringe effects which arise from the subsidiary rings of the Airy 
discs. 

5.3. Abbe's theory of microscopic vision 

The problems that we have discussed in the last two sections worried 
the microscopists of the 19th century very considerably and it was not 






120 



IMAGES 




Fig. 5.10. (a), (b), (c) and (d) Four recombinations of successively reduced 
portions of the optical transform of a transparent letter H. 

until the German physicist Abbe produced his famous theory that the 
behaviour near the resolution limit began to be understood. The 
convolution approach described in the last sections has really super- 
seded Abbe's ideas but, partly for historical reasons, and partly because 
of the elegance and simplicity of the idea-which remains correct 
though somewhat limited in application— we shall devote a little 

space to it. ,.•«■*• 

Abbe began by considering just how the image of a regular diffraction 
gratine of parallel rulings might be produced in a microscope. 
Fig 5~1 1 shows the kind of ray diagram with which his arguments have 
been illustrated. We show only the objective of the microscope-the 
eyepiece has little effect on the resolution problem. The real image ot 
the grating G appears at G'. But any set of parallel rays from the 
grating slits will come to a focus in the back focal plane of the objective 
before going on to form the image. The back focal plane will thus 






PERTURBATIONS OF THE IMAGE 



121 



contain the orders of diffraction of the grating. The diagram shows 
very clearly how the scattering pattern in the back focal plane then 
goes on to become the image. By means of a series of extremely 
elegant experiments in which Abbe actually cut out various orders of 
diffraction in the back focal plane of his microscope objective, he was 
able to study the effects on the image — rather as we did in a differ- 
ent way for figs. 5.1 and 5.2. 




Fig. 5.11. Ray diagram illustrating Abbe's theory of the microscope. 
G is the regular grating, L the microscope objective, F the orders of diffrac- 
tion in the back focal plane of L and G' is the image of G. 



On the basis of his experimental evidence he formulated the principle 
that the final image in a microscope will not be an exact replica of the 
object but will be the replica of an object which would have as its 
diffraction pattern that portion of the diffraction pattern that is allowed 
to enter the image-forming system. Thus if we take fig. 5.2 (a) as an 
example: the image obtained by letting this into an image-forming 
system will be of an object which has 5.2 (a) as its total diffraction 
pattern. And this is precisely what 5.2 (b) is. For a regular object, 
the periodicity in the image will disappear when only the zero order of 
the diffraction pattern is allowed to contribute to the image. 

Abbe recognized that his theory applied to irregular as well as to 
regular objects and really laid all the foundations for modern image- 
processing techniques — which we shall discuss in more detail in the 



122 



IMAGES 



PERTURBATIONS OF THE IMAGE 



123 



last chapter. As I said at the beginning of this section, his theory is 
largely superseded by the Fourier transform-convolution approach 
bur it is very good practice to think through some of the examples 
we have given in terms of his theory. Indeed one of the delights of 
this kind of optics is that very often there are several different approaches 
that can lead to a given result. You begin to have a much more 
confident feeling about a subject if you can arrive at the same result 
by more than one method. 

5.4. Image * formants'— the fingerprint of the apparatus 
In the study of sound there is a very important principle that each 
time a sound wave passes through any kind of system, the system will 
impose its own fingerprint or formant characteristic on the sound. 
For example, our voices are created first by the vibration of the vocal 
cords. The rather harsh buzzing sound then passes through the 
throat-nose-mouth system with all its cavities and at least two distinct 
formant characteristics are imposed on it. There is a formant charac- 
teristic for each vowel sound, and this is the same whoever makes the 
sound, and there is a formant characteristic that distinguishes one 
person from another. A further example would be the curious things 
that happen to the sounds created in a concert hall when they are 
transmitted by radio. The differences between the sound reproduced 
by a super hi-fi system, a moderately good portable radio and a 
miniature transistor radio are considerable. In each case the receiver 
has imposed its own formant characteristics. Some of you will have 
tried to build amplifiers and will know how difficult it is to produce 
uniform amplification at all frequencies, and it is the departures from 
uniformity of amplification that we call the formant characteristic. 

The parallel with what happens in image-forming systems is remark- 
ably close. The idea of diffraction patterns being made up of sum- 
mations of fringes has been mentioned several times and we have also 
talked about the reciprocal relationship between the spacing of two 
points of an object and the spacing of the corresponding fringes in 
the diffraction pattern. In terms of Abbe's ideas, the finer the detail 
on the object, the further from the axis would be the corresponding 
order of diffraction, and it is clearly these higher orders that are likely 
to be cut out in an imaging system. Thus it is likely to be the high 
spatial frequency element of the pattern that will be the first to go- 
as indeed we saw in the case of Mickey Mouse in fig. 5.9. The 
elimination of spatial frequencies above a certain level would be a 
simple kind of formant that would occur wherever an image is formed 
by a system with a limited aperture. But the effect is much more 






general and we find that all kinds of variations in the frequency content 
can occur on the way through a system and each element will impress 
its own formant characteristic on the image. Thus if we are concerned 
with the transmission of a television picture, it must now be clear that 
electronic distortions or modifications of the signal between trans- 
mitter and receiver, or in the receiver itself, will change the picture. 
Indeed, we can generalize Abbe's theory and say that the final picture 
will be of an object that would have produced the signal that is finally 
fed to the picture-forming circuits rather than of the object itself. 
Another example of a way in which a formant might arise is in any 
system in which scanning is involved. If the spot, instead of being 
small, becomes larger, or becomes elongated, or indeed any shape 
other than a mathematical point, then the image will be the convolution 
of the true image with this shape. This, in Fourier transform terms, 
corresponds to multiplying the scattering pattern by the diffraction 
pattern of the deformed spot and so the result is just the same as for 
a microscope or telescope with an odd-shaped aperture. 

It may seem that this section is becoming somewhat philosophical, 
but the principles being expressed are of great importance. Perhaps 
they will become clearer in the last chapter, when we shall look at a 
great many examples of how our ideas work out in practice. It is 
particularly important to try to understand the principles in order to 
study various systems for image correction or image enhancement, 
■aich as are used for example in processing television images returned 
from spacecraft. 

So far we have talked about image-forming systems in which we 
assumed the lenses to be free of all aberrations. In fact, lenses are 
rarely free of aberrations. They may for example suffer from spherical 
aberration — which means that a spherical wave incident on the lens 
does not merely change its curvature but ceases to be spherical after 
passing through the lens. It cannot therefore come to a point focus 
and any image produced by it will be distorted. There are other 
aberrations that can arise which we will not consider in any detail. 
It will suffice to say that all can be regarded as formants of the system 
and, in some cases, we can correct the defects by compensating for the 
unwanted formants. 

5.5. Methods of measuring the performance of imaging systems 

In modern optics it has become customary to specify the overall 
performance of an imaging system in one of two ways. It becomes 
necessary to specify it because, if we are producing an image of some- 
thing we have never seen before (e.g. a new strain of bacteria) we 



124 



IMAGES 



PERTURBATIONS Ol THE IMAGE 



125 



cannot compare the image with the object because we do not know 
what it should look like. We therefore need some kind of objective 

criteria. 

One of the standard methods is closely related to the formant idea 
described in the last section. We consider each spatial frequency in 
turn and find out how well the system reproduces it. The result is a 
function called the transfer function; it describes how each frequency 
is transferred through the system and what happens to it when it 
reaches the other side. In a system free from all aberrations (a so- 
called diffraction-limited system) the transfer function would simply 
be of unit magnitude up to the maximum spatial frequency admitted 
to the system and zero outside this. For systems with aberrations 
the function becomes more complicated and may involve phase as well 
as amplitude changes. 

The second standard method is to use what is called the point 
spread function. This is simply the image on one side of the system 
resulting from a mathematical point source on the other side. Again 
for a diffraction-limited system the result is quite simple: the point 
spread function is simply an Airy disc. For systems with aberrations 
it might be quite a complicated shape and may vary from point to 
point of the object. In Section 6.9 the method of studying electron 
microscopy essentially involves the experimental determination of this 
point function and fig. 6.28 shows the complex form that it can take 
when aberrations are present. 

5.6. Depth of focus problems 

There is one image defect that is worth picking out from the rest for 
special consideration before we proceed to practical applications, and 
that is a manifestation of the focusing problem which has already 
cropped up once or twice. We have tended to think of two-dimensional 
images but often they are produced from three-dimensional objects. 
Simple geometry shows that it is unlikely that we can devise a lens that 
will produce a perfectly focused two-dimensional image of a three- 
dimensional object, and this supposition is amply borne out in practice. 
Fig. 5.12 is a series of photographs which illustrate the phenomenon. 
We start with a photograph of a young lady and successively move 
closer until in 5.12(c) we have just about reached the limit with an 
ordinary camera; the origin of the effect is discussed in section 6.3. 
It is very obvious that only one plane in the texture of the fine wool 
scarf is being reproduced satisfactorily. We then transfer to the 
microscope and in 5.12 (/) and (/?) increasing magnification shows finer 
and finer detail and the problem of depth of focus remains with us. 



Is there a solution? There is, but it involves completely different 
imaging system which depends on scanning rather than on a simul- 
taneous transformation using lenses. We have illustrated the effect 
here by transferring to the scanning electron microscope and in figs. 
5.12(g), (Oi (./') and (k) greater and greater detail is shown without any 
depth of focus problems. The flakes that can be seen in the pictures 
at higher magnification are odd specks of face powder on the scarf! 
How does the scanning system solve the depth of focus problem? 
The point is simply that the scanning beam is a fine pencil of electrons 
(or of light in the flying-spot microscope) which traverses the specimen 
and at any one moment the detector is receiving information only 





(a) 



ih) 








(c) 



(</) 



Fig. 5.12 (cont. overleaf). 



126 



IMAGES 








(e) 



(f 





(*) 



m 





(0 en 

Fig. 5.12 {com. opposite), 



PERTURBATIONS OF THE IMAGE 



127 




Fig. 5.12. (a)-(e) Photographs with an ordinary camera at closer and closer 
range. (/) Optical micrograph approx. x 25. (g) Scanning electron micro- 
graph approx. x 25. (//) Optical micrograph approx. x 100. (/) Scanning 
electron micrograph approx. x 100. (/) Scanning electron micrograph 
approx. x 600. (Ar) Scanning electron micrograph approx. x 6000. The 
pale flakes in (/), (j) and (k) are traces of face powder. The regular scales 
are well-known features of wool fibres. 







Fig. 5.13. Principle of the scanning microscope: if the incident beam of 
radiation from G. the gun, is near-parallel the scattering from the object is 
not affected by the variations in height of the specimen S. 

about that one point. It does not matter whether the points scanned 
are all in one plane or not provided that the pencil of radiation is a 
parallel one. Fig. 5.13 is an attempt to illustrate this point. We shall 
consider some more of the practical consequences in the various 
sections of the final chapter. 



128 



IMAGES 



5.7. Conclusion 

In the five chapters so far, we have tried to lay the foundations for 
imaging theory. We have shown that in all imaging processes there 
are the two stages, the first being either scattering or radiation and 
the second recombination or focusing, and we have seen that this 
basic view of image formation applies to all kinds of radiation. We 
have seen that the scattering process itself can be studied in detail and 
sometimes indeed is the only aspect that can be studied. We saw for 
example that for X-ray studies of crystals it is impossible to complete 
the recombination process experimentally and we have to resort to 
alternative techniques. We have also seen that there are various 
different ways of performing the recombination stage, some of which, 
such as using a lens, enable us to do the whole thing simultaneously 
whereas others, such as scanning, involve point-by-point reconstruction. 
Finally we saw that all image-forming systems impose their own 
characteristics on the image that is produced and if we are to interpret 
our images properly we really should know as much as possible about 
our image-forming system. On the basis of all these ideas we shall 
now spend some time looking at a variety of practical image-forming 
systems to see how the principles already established work out in 
practice. 






6. Applications and results 



" Merely corroborative detail, intended to give artistic 
verisimilitude to an otherwise bald and unconvincing narra- 
tive ". 

W. S. Gilbert 

The Mikado, Act II 



6. 1 . Introduction 

I have tried in the earlier chapters of this book to demonstrate that 
every system used to form an image conforms to the same series of 
basic physical principles. The illustrations I have used in the course 
of the book have been fairly general and there may have been numerous 
occasions when you wished for a little more detail or elaboration. In 
this last chapter, I propose to give practical details and descriptions for 
a considerable number of image-forming systems — and indeed in 
some cases systems that might not normally be considered as image- 
forming. I hope that they will not only prove interesting for their own 
sake but will also help you to see how the general principles developed 
in Chapters 1 to 5 work out in real life. It has always been one of my 
firm beliefs that physics should not be presented merely as a series of 
abstractions and generalizations. Though these clearly have a place, 
it is of great importance that they should be set in the context of real 
applications. This chapter therefore represents my attempt to be true 
to this belief. Inevitably it will be somewhat disjointed because it is 
really a collection of separate short descriptions. It also represents 
a personal selection which is far from being complete. 

6.2. The eye 

The eye is, of course, the most familiar of all image-forming systems 
and, indeed, we started the discussions in Chapter I with a considera- 
tion of the process of seeing. Fig. 6. 1 is a standard diagram of a 
cross-section of a human eye. From our standpoint, the eye is a device 
which takes a sample of the pattern of light radiated or scattered by 
objects around us and, by means of a combination of phase adjusters, 
produces a representation on the retina which is then converted into the 



130 



IMAGES 




Cornea 



Fig. 6.1. Diagrammatic longitudinal section of an eye. 

sensation of vision through a chain of electrical impulses to the brain. 
The phase adjusters are the combination of the curved surfaces of the 
cornea, the ' humours ', or liquids which fill the eye, and the crystalline 
lens itself. The latter can be changed in shape by the ciliary muscle 
to vary the adjustment and hence to focus on either near or far objects. 
This process, known as accommodation, usually deteriorates with age. 
An ordinary converging lens with spherical surfaces suffers, among 
other things, from the defect known as spherical aberration. The 
centre zone of the lens near to the axis has less phase-adjusting effect 
(By this I mean effect on the curvature of the incident wave, not 
the absolute phase shift which, of course, is greater at the centre where 
the lens is thicker) than the zones nearer the periphery, and the focal 
length for the outer parts is shorter than that for the inner. This can 
be demonstrated by forming an image of a distant lamp by a simple 
lens covered with a card with a small hole allowing light to pass only 
through the central zone of the lens. If the card is now replaced with 
one having an annular hole allowing light to pass only through the 
outer edges, refocusing is necessary and the outer part is easily shown 
to have a shorter focal length. The experiment works most effectively 
with a near-monochromatic source and a short focus lens. 

The combined lenses of the eye also exhibit spherical aberration, but 
it is difficult to demonstrate this convincingly on a live eye because the 
effect is masked by other effects. In particular the increased depth of 
focus experienced when a small aperture is placed in front of the eye 
(discussed in the next section in connection with cameras) and the 
automatic focusing action of the eye lens create complications. 

The eye also suffers from chromatic aberration and this can be 
demonstrated by observing a distant bright source of light through a 



APPLICATIONS AND RESULTS 



131 






piece of deep violet glass or gelatine which cuts out most of the centre 
region of the spectrum and passes only the red at one end and the 
violet at the other. Your eye will usually focus on the red image and 
this will be surrounded by a blurred violet image. Opticians sometimes 
use the effect as a ' fine adjustment ' in sight testing. A test chart is 
illuminated partly in red light, partly in green and partly in blue. 
When the patient sees the green part clearly and the red and blue parts 
are equally blurred, the correcting lenses are of the optimum focal 
length. 

The final feature that is important from out point of view — though 
of course a whole book could be written about the eye and its quite 
remarkable properties and powers— is the beautiful economy of design 
of the eye. 

If the eye is focused on a point source of light a long way away, we 
have exactly the conditions for producing Fraunhofer diffraction. 
The pupil of the eye will then give rise to an Airy disc around the 
image of the point source. The same effect will, of course, occur 
whatever the object being viewed; each part of the object will be repro- 
duced on the retina as an Airy disc (compare fig. 5.3) and the image on 
the retina is a convolution of a true image and the Airy disc correspond- 
ing to the pupil. 

Now the pupil varies in size from I to 6 mm according to the level 
of illumination. If we take 2 mm as the diameter in average daylight 
conditions, the angular diameter of the central disc on the retina will be 
just over 2 minutes of arc for a wavelength (measured inside the eye) 
of 5 x 10 ' in. The focal length of an average eye is about 1-5 x 10 - ' 2 m 
and so the kind of detail which is just on the limit of resolution is about 
4x 10" m. At the most sensitive area of the retina, the density of 
detecting cells is such that they are about 2x 10 ,: m apart. In other 
words the limitation on resolution provided by the structure of the 
retina is just a little better than the average limit set by diffraction at the 
pupil, a remarkable example of the economy arrived at during the 
process of evolution. 

6.3. The camera 

Many years ago when I was a young schoolboy I became interested in 
photography and a man I knew who was a keen amateur photographer 
offered to show me how to develop films and to make prints. He had 
a dark room fixed up in a cellar and I can still remember the astonish- 
ment I felt when he put the light out and there on the white-washed 
wall was an inverted image of a tree with its branches moving in the 
breeze — all in colour! My friend was annoyed and searched for the 



132 



IMAGES 








m 

Fig. 6.2 (cont. opposite). 



APPLICATIONS AND RESULTS 



133 




(0 
Fig. 6.2. (a) Principle of the Camera Obscura. (b) The Camera Obscura 
at Dumfries, (c) Inside the Camera Obscura at Dumfries. (Photo by 
David Hope, Dumfries.) 



hole in the blind covering the window that was producing this unwanted 
image. It was of course a primitive pin-hole camera. I think the 
fascination that images have always held for me dates from that 
moment. Sometime later I visited a sea-side resort where a small hut 
on the cliffs advertised a ' Camera Obscura '. I duly paid my penny 
(old type) admission and was again enthralled. (Remember that this 
was before the days of television or colour films). There on the white 
table in the middle of the hut was displayed in full colour and, of course, 
complete with movement, a panorama of the beach and sea front. In 
this case a large lens in the roof was forming the image with a plane 
mirror to throw the image down on to the table. The system is just an 
enlarged version of the eye — though without the remarkable crystalline 
lens. Focusing was done simply by adjusting the position of the lens 
relative to the mirror and hence to the table (fig. 6.2 (a)). 

During a recent holiday in Dumfries, I was delighted to find that 
the Camera Obscura at the Burgh Museum is still operating. It was 



134 



IMAGES 



APPLICATIONS AND RESULTS 



135 



set up in 1835 at a cost of £27.105. It is believed to be the oldest 
Camera Obscura in the world that is still operational. The Old Mill 
which houses it can be seen in fig. 6.2 (b) and a group of visitors 
viewing the image on a concave table can be seen in fig. 6.2 (c). 

The camera is a portable version of this device and it is important to 
remember that the camera focused on infinity is also a device for 
producing Fraunhofer diffraction patterns and that the image will be a 
convolution of the true image with the Airy disc of the lens. When 
focused on any other distance, the effect will be similar, as then the 
Fresnel pattern will resemble the Airy disc. A typical 35 mm camera 
(i.e. camera taking pictures on film 35 mm wide) might have a lens 
of focal length 5xl0- 2 m. The diameters of camera lenses are 
specified by so-called F numbers. This number is defined as the 
ratio of the focal length (/) to the effective diameter of the lens (d). 
Thus an F number of 8 means that the focal length is 8 times the effec- 
tive diameter of the lens (I use the term ' effective ' because in a complex 
lens with many components it may not correspond to any actual 
component diameter but is the diameter of a thin single lens that would 
be equivalent). Now the magnification— if the lens is set to focus on 
a fairly distant object— is approximately the ratio of focal length to 
the distance away from the object ( /) and so the area of the image of a 
particular object is proportional to/ 2 // 2 . The amount of light entering 
the lens is proportional to its area which in turn is proportional to d 2 . 
Thus the image intensity (the amount of light per unit area) is propor- 
tional to d 2 +f 2 ;l 2 which is simply I 2 !F 2 and so for a given object distance 
the intensity of the image is proportional to 1 ;F 2 . Thus, if we choose 
an F number twice as large, the intensity goes down by a factor of 4. 
(Thus the exposure at F16 is 4 times that needed at F8.) Fig. 6.3 shows 
how lenses of different diameters and focal lengths can have the same 
F numbers. 

The typical camera with a lens of 5 x 10~ 2 m focal length operating 
at F2-5, would have a lens diameter at 2x lO" 2 m, and would produce 




a central disc with light of wavelength 5x I0~ 7 m of angular diameter 
about 12 seconds of arc. Its diameter on the film would be about 
3 x 10~ 6 m ( = 3 micrometres). Clearly this is not going to produce 
any serious problems unless the negative is to be enlarged to a very 
large size. The problem of the size of the individual grains of silver 
in the photographic image is likely to become significant before the 
lack of resolution of the lens unless very special fine-grain films are 
used. Fig. 6.4 shows examples of very highly enlarged negatives with 
both very fast (large grain) and very slow (fine grain) films. 



Fig. 6.3. Three lenses of different focal length but with the same F number. 





■r— — ' - —r- 

H 


. 



(a) 



(b) 



Fig. 6.4. (a) Micrograph X 200 of white, grey and black lines photographed 
with the fastest film available for amateur photography, (b) Micrograph 
< 200 of the same white, grey and black lines as for (a) but photographed on 
a very fine grain film for amateur use. 

There are many fascinating problems that we could discuss in relation 
to cameras, but I shall pick out just two which link very closely with 
topics discussed in earlier chapters. The first concerns the depth-of- 
focus problem (see Section 5.6). The series of photographs of fig. 5.12 
illustrates clearly the problem of reproducing a three-dimensional 
object on a two-dimensional film. In Chapter 5 we saw how scanning 
could overcome the difficulty and in section 6.9 we shall consider it in 
more detail. 

There is, however, another important point to be made, and that is 
that the depth of focus is dependent on the lens aperture. This is 
most easily understood by a purely geometrical approach in the first 
instance. Fig. 6.5 (a) shows a convergent wave coming to a point 



136 



IMAGES 





Fig. 6.5. Size of ' out-of-focus ' patch (a) for lens with large F number. 
(/>) for lens with much smaller F number. 

focus at P and then diverging. At P' and P" the point would become 
a patch of a diameter corresponding to the diameter of the cone at these 
points. If we now consider fig. 6.5 (b) we have a converging wave of 
solid angle such as would be produced by a lens of lower F number. 
Now if we move to P' or P" the diameter of the patch is much less than 

in fig. 6.5 (a). 

Suppose we are photographing a three-dimensional scene and the 
lens is adjusted so that a particular plane in the scene is focused sharply 
on the film. As in earlier examples we can imagine the scene to be 
made up of mathematical points and, when the image is sharply focused, 
each of these (if we ignore the Airy disc problem) is reproduced as a 
mathematical point. If we now transfer our attention to a plane at 
some other distance from the lens which focuses at a point in front of 
or behind the film, then each point on the object will be reproduced as 
a circle as at P" in fig. 6.5 (a) or (b). The image will now be a convolu- 
tion of the true image with the circular patch. 

It should now be clear from figs. 6.5 {a) and (b) that for a given plane 
on the object the patch size will be greater, and the blurring of the 
image consequently also greater, the larger the aperture of the lens; or 
putting it the other way round, if the eye can tolerate blurring up to a 
certain patch diameter, planes in the object of greater ' front-to-back ' 
depth will appear ' in focus ' if the aperture is smaller than if it is larger. 
Fig. 6.6 shows a series of photographs of the same object from the 
same position and with the same lens but with varying apertures. (The 
exposures were adjusted to compensate for the varying amounts of 
light admitted.) 



APPLICATIONS AND RESULTS 



137 




(«) 



(6) 









B * 






■ * 






^K * 






^K * 






^K * 






^K * 






^K * 






I 


i 




■' ' 






I " 






^E c 




BI 




K * 


* 1 


R* 








1 aBH^n 


W»j, 






gfl 




i ? ■ 




1 : " - 








^F-~ 






^& 




■f %■ 






1 


•• - 




1 






1 






1 JC 






■ " 






W l ' 






(c) 



(d) 






Fig. 6.6. Photographs of a ruler with the same camera and lens from the 
same position and in each case focused on the 100 mm mark. The exposures 
were adjusted to give similar contrast, (a) F 1-8; (b) F4;(c)F8; (d) F 16. 
K 



138 



IMAGES 



Thus we have the paradoxical situation that in order to increase 
depth of field we need to use a small aperture but in order to avoid 
diffraction problems we need a large aperture. In practice, as has 
already been mentioned, the diffraction problem is rarely of great 
consequence in normal photography. 

The final point I want to mention here, because it links well with 
our other discussions, concerns trick effects that are sometimes used 
with cameras— particularly cine cameras. We have already used the 
notion that a camera focused on a point source is a suitable device for 
obtaining Fraunhofer diffraction patterns, and indeed we saw in section 
5.1 that a point source would produce the Airy disc of a lens aperture. 
If some kind of mask made up of opaque and transparent areas is 
placed over the lens, then the lens aperture is being multiplied by a 
function represented by the pattern of the mask. The image must then 
consist of the convolution of the true image with the diffraction pattern 
of the patterned mask. Fig. 6.7 (a) shows what happens when a scene 
containing several bright lights is photographed with fine silk gauze 
over the lens. Since the centre spot of the diffraction pattern of the 







(a) 
Fig. 6.7 (com. opposite). 



APPLICATIONS AND RESULTS 



139 




;---: ; i : ;::::;:::::-; 



(<") 






Fig. 6.7. (a) Scene photographed with gauze over lens of camera, 
for (a) but with gauze rotated, (c) Close up view of gauze. 



(b) As 



140 



IMAGES 



gauze is so much brighter than everything else, the convolution with the 
background scene reproduces the scene fairly well but where there is a 
bright light the full diffraction pattern stands out. Rotating the gauze 
in its own plane leads to a rotation of each of the diffraction patterns 
about its own centre, as shown in fig. 6.7 (b). This is a very good 
additional illustration of the convolution-multiplication property. 

6.4. Optical telescopes 

Telescopes are often presented as devices for producing enlarged 
images of distant objects but, as will have been realized from earlier 
chapters, the enlargement is only part of the story; the increased 
light-gathering power above that of the pupil of the eye can be every 
bit as important. The usual diagrams of the two simplest telescope 
systems with lenses emphasize the light-gathering feature more than 
the magnification. Fig. 6.8 (a) shows the Galilean telescope (sometimes 
called a terrestrial telescope as it does not invert the image) and 6.8 (b) 
shows the Keplerian telescope (sometimes called the astronomical 
telescope because it inverts the image but this is of no consequence in 
viewing stars). In each a parallel beam of considerable cross-section 
enters the objective and emerges still parallel but very considerably 
reduced in cross-sectional area; the intensity of the image is thus 
considerably increased. Simple telescopes of this type do not give 
images that are free from aberrations, and many variations using 
compound lenses for the objective to correct for both chromatic and 
spherical aberrations and various other errors are known; a great 
variety of different eye-pieces, most of which correct at least to some 
extent various aberrations, may also be incorporated. 

Because of the difficulty of casting very large lens blanks, the problem 
of supporting them without interfering with the optical properties and 
the complexity of correcting the various aberrations when the aperture 




1 



(b) 

Fig 6.8. (a) Schematic diagram of Galilean telescope in normal adjustment. 
(b) Schematic diagram of Keplerian telescope in normal adjustment (often 
simply described as ' the astronomical telescope ')• 















APPLICATIONS AND RESULTS 



141 



is large, most really large telescopes are made using a mirror as the 
objective. A spherical mirror reflects all wavelengths in the same way 
and hence is inherently free from chromatic aberration. If the mirror 
is made paraboloidal instead of spherical, the spherical aberration 
problem disappears and a parallel beam of light covering the whole 
aperture of a large mirror can be brought to a point focus. 

Fig. 6.9 shows the Newtonian telescope arrangement. The 
paraboloid collects the light and the image is reflected through the 
side wall by a small mirror, known as the ' diagonal ', to an eyepiece. 




Fig. 6.9. Schematic diagram of Newtonian reflecting telescope in normal 
adjustment. 

What is the effect on the image of interposing this mirror and its 
support? We can approach this problem using our knowledge of 
diffraction and convolution. The system gives, for each object point, 
a figure which is the diffraction pattern of the aperture; the aperture, 
in this case, is a large circular hole with a relatively small opaque spot 
at the centre. Optically, the combination may be thought of as a small 
circular hole subtracted from a larger one! Each hole has as its 
diffraction pattern an Airy disc and we need to subtract the one from 
the other. Fig. 6. 10 (a) shows the distribution in amplitude of the Airy 
disc from the large aperture and (b) shows that from the small aperture 
assumed to be one tenth the diameter of the large one; (c) shows their 
difference. It is clear that, although the fringe pattern is modified 
slightly, the predominant effect— the central peak— is very little 



142 



IMAGES 

amplitude 
/l\ 



(«) 



-A 



(b) 



(c) 



-x 



A 



amplitude 






amplitude 



f- 



Fig. 6.10. (a) Amplitude distribution in Airy disc pattern of full aperture. 
(b) Amplitude distribution in Airy disc pattern of an aperture of the same 
size and shape as the diagonal mirror, (c) (a) minus (b), the amplitude 
distribution in the diffraction pattern of the annulus which is the effective 
aperture of the Newtonian telescope. 

different from that due to the large aperture on its own. And this 
really answers the question: the convolution of any image with the 
function of 6.10 (a) will not differ very much from its convolution with 
6.10 (c) and hence the diagonal does not upset the image. 

Again there are many variants on this system, including the Hale 
type of telescope in which the observer himself is seated inside the 
telescope near the focal point— though clearly this can only apply to 
pretty large instruments such as the 200 inch Mount Palomar telescope. 
Whatever kind of telescope is being used, however, the basic function is 
to pick up as big a fraction as possible of the scattering pattern pro- 
duced by the object under examination and to recombine it into an 
image; a major part of the problem of telescope design is to ensure that 
this recombination is done with as little error as possible. One of the 
principle points of interest, therefore, is the method of testing the 
spherical or paraboloidal mirrors developed for telescopes, partly 
because the successful operation of telescopes depends critically on the 



APPLICATIONS AND RESULTS 

success of this testing, and partly because the test itself relates very 
directly to our discussions on image formation in the earlier chapters. 
Let us first consider a method of testing a spherical mirror and let us 
further assume to begin with that our mirror is perfect. We set up a 
tiny point source of light and, placing it next to one eye, move both the 
eye and the point source around until we have the source and eye very 
close to the centre of curvature of the mirror (fig. 6. 1 1 (a)). In terms 
of image formation the mirror is being illuminated and its scattering 
pattern is the returned image of the point source — the Airy disc of the 
aperture. The eye recombines this to give an image which shows 
completely uniform brightness over the surface of the mirror; it is this 
uniform brightness that signifies to the observer that his eye and the 
source are close to the centre of curvature. 



Knife edge 




knife edge 



knife edge 



knife edge 





(b) 



(c) 



(d) 






Fig. 6.11. The Foucault "knife-edge' test for spherical mirror, (a) The 
experimental arrangement, (b) Knife edge nearer to mirror, (c) Knife 
edge nearer to eye. (d) Knife edge at centre of curvature. 

Now a knife edge, often a razor blade, is introduced in front of the 
eye and is passed (say) from left to right across the field of view. 
Simple geometry (fig. 6.11) shows that, if in fact the blade is nearer 
to the mirror than its centre of curvature, the eye will be conscious of a 
shadow moving from right to left (b). If the blade is nearer to the eye 
than the centre of curvature of the mirror, the shadow will move with 
the blade from left to right (c). If the blade is exactly at the centre of 
curvature the whole aperture of the mirror will appear to go dark 
instantly (d). 

Now let us suppose that there are some small errors in the mirror — 
perhaps odd scratches, patches of dirt or departures from a perfectly 



144 



IMAGES 



spherical shape. Without the blade in position, unless the effects are 
really gross, they will not be seen by the eye. However, if the blade is 
brought across at the centre of curvature until the illumination of the 
whole mirror is just extinguished, all the defects will suddenly appear 
quite clearly. Why is this? 

The defects can be thought of as objects across the aperture of a 
system producing an approximation to the Fraunhofer diffraction 
pattern and so the image formed near the eye will no longer be a simple 
Airy disc but may be quite a complex pattern. If the defects are smaller, 
as is inevitable, than the aperture of the mirror, then, because of the 
now familiar reciprocal relationship between objects and their diffrac- 
tion patterns, the diffraction pattern will be larger than the Airy disc. 
Consequently when the blade has advanced to cut off the Airy disc 
and hence the main illumination of the mirror, the larger parts of the 
diffraction pattern will still be available to the eye for recombination. 
Following Abbe's principle, the eye will recombine the pattern to form 
an image of something that would give just that as its diffraction pattern 
without the Airy disc; in other words it produces an image of the errors 
without the overpowering overall illumination of the whole surface. 
This test, known as the Foucault knife-edge test, forms the basis 
(though a great many complicated variations have been developed) of 
most methods of examining the perfection of mirrors. It is also very 
closely related to the Schlieren technique to be described in Section 6.6. 

A paraboloidal mirror has to be tested in a slightly different way, 
with an auxiliary optically flat mirror. Fig. 6.12 shows the arrange- 
ment; the eye and source are now at the focus of the paraboloid, and 
hence a parallel beam is produced which is then returned by the flat 
and hence to the focal point. The effect of errors or imperfections can 
be understood in exactly the same way as for the spherical mirror. The 
provision of a suitable point source sometimes presents difficulty and 
an excellent solution is to use a highly polished steel ball to create a 





Fig. 6.12. Adaptation of knife-edge test for paraboloidal mirror. 






APPLICATIONS AND RESULTS 



145 



small image of a bright lamp placed some distance away to one side. 
Such a source— i.e. the steel ball— can be placed very closely to the eye 
without discomfort. 

The process of shaping or testing a paraboloidal mirror is a fascinating 
one but is also quite difficult to achieve with high precision, and conse- 
quently a number of alternatives have been developed. Perhaps the 
most famous of these is the Schmidt system, which is widely used for 
medium size astronomical telescopes for photographic purposes (in the 
range up to 2 m diameter). Again it is picked out here because it 
provides a beautiful illustration of some of the principles developed in 
earlier chapters. 

A spherical surface is relatively easy to produce. Experience shows 
that if two pieces of glass are rubbed together with grinding powder 
between and are frequently rotated relative to each other, one becomes 
concave and the other convex and both acquire spherical surfaces to 
quite a high degree of precision. Indeed spherical surfaces are the 
only possible ones (other than absolutely flat surfaces) which can slide 
over each other in such a way that all points remain in contact. If the 
surface were other than spherical the high points would be subject to 
grinding and the low points protected and the surfaces would gradually, 
and more or less automatically, become spherical. 

The Schmidt system starts with a spherical mirror and then attempts 
to correct for the difference between the required paraboloidal surface 
and the given sphere. Figs. 6.13 (a) and {b) show the difference 
between the behaviour of a paraboloid and a sphere. The waves 
incident near the centre of the sphere come to a focus too far away 
from the mirror and those near the periphery come to a focus too near 
to the mirror. The Schmidt system involves adding a so-called 
corrector plate which is a sheet of transparent material of thickness 
which is different at different distances from the centre. The introduc- 
tion of this plate effectively multiplies the amplitude distribution of the 
wave front by the transmission function of the plate. But since it is 
almost completely transparent the magnitude is unaffected and it is 
only the relative phases of the waves that are affected. The device can 
be thought of as introducing a ' formant ' to the system (see Section 
5.4). A cross-section of a corrector plate is shown in fig. 6.13(c). 
It can be seen that near the centre the added phase difference diminishes 
as we move out from the centre; the plate therefore acts like a conver- 
ging lens and this in turn moves the focal point of these waves nearer 
to the mirror. Towards the periphery, the plate thickens again and so 
in the outer region it acts like an annular diverging lens and so moves the 
focal point of the peripheral waves away from the mirror. If the 



146 



IMAGES 



dimensions of the plate are carefully chosen, all the waves arrive at the 
same focal point and the combined plate and spherical mirror behave 
like a paraboloidal mirror. A fuller mathematical investigation 
shows that if the plate is placed in the plane of the centre of curvature 
of the mirror other aberrations are diminished as well and the system 
behaves better even than a paraboloid. Fig. 6.13 (d) is a view of a 
building with an ordinary camera; fig. 6.13(e) is a photograph of a 
detail of the building from exactly the same position (90 m from the 
building) using a small (0-12 m) Schmidt system. The remarkable 
reproduction of detail testifies to the quality of the optical system. 

Two final points in our consideration of optical telescopes will now 
be mentioned. Both involve restricting the aperture in certain ways 
in order to perform measurements. The first permits the measure- 
ment of the diameter of a star and is the so-called Michelson stellar 
interferometer. In its simplest form it consists of two slits, whose 
separation can be varied, placed over the aperture of the telescope. 
Following our usual practice we can think of this double slit as a 
function multiplying the aperture and hence instead of the usual Airy 
disc pattern, each star image will be convoluted with the diffraction 
pattern of the double slit. The diffraction pattern of the double slit 
is of course a normal Young's fringe pattern and so each star image now 
has fringes across it as well. If the spacing of the slits is increased the 
spacing of the fringes is decreased but the fringes will remain visible 
only if the illumination of the slits is coherent. Now the degree of 
coherence of the illumination of the telescope aperture depends on the 
size of the star source. The bigger the source the lower the coherence 
and so the more quickly will the fringes disappear as the slit spacing 
increases. A fairly simple mathematical relationship permits the 
separation of slits for which the fringes disappear to be correlated to the 
star diameter. The experiment is precisely the same in principle as that 
described in Section 2.8 for the measurement of spatial coherence. 

For small stars, the aperture of the telescope may not be big enough 
to allow the slits to be moved sufficiently far apart. A framework 
carrying moving mirrors (fig. 6.14) can be added to the telescope to 
increase the effective aperture for the purposes of the measurement. 

The other modification of the aperture that is of interest is the tech- 
nique known as apodization. A telescope of given diameter cannot 

Opposite: 

Fig. 6.13. (a) Parallel beam incident on paraboloidal mirror, (b) Parallel 
beam incident on spherical mirror, (c) Schmidt plate which will make 
(b) behave as (a), (cl) View of building with ordinary camera, (e) View from 
same position with 012 m Schmidt camera. 




148 



IMAGES 



resolve two stars closer together than a certain angular separation and 
it would seem that the only way to improve the resolution would be to 
increase the aperture — usually an impractical solution. Paradoxically 
it can be shown that by blocking up part of tfie aperture some improve- 
ment can result! A circular opaque disc is added to leave an annular 
ring of the lens or mirror as the only part contributing to the image. 





^ 


1 

1 


' 


Effective \z ~*" 
slit separation <$ _*. 






4 




> i 





n 



Fig. 6.14. Principle of the Michelson stellar interferometer. 

We can again understand how it works if we remember that the 
diffraction pattern of this annular ring will be reproduced by convolu- 
tion at each star image. Fig. 6.15 is a plot of the Airy discs of the 
whole aperture of the telescope and of an aperture the size of the opaque 
disc. The diffraction pattern of the annulus is the difference of the 
two, which is shown on the same figure as the thick line. The impor- 
tant point to notice is that the central peak of the diffraction pattern 
of the annulus has a smaller diameter than that of the full aperture. 
Hence the resolution will be increased. 

But this seems like something for nothing! We seem to be gaining 
information by throwing information away. The resolution of the 
paradox is also contained in fig. 6. 15 (and in fig. 2.12(/) which shows 
the actual diffraction pattern of an annulus). You will notice that the 
rings surrounding the central peak for the pattern of the annulus are 
much more intense than those for the plain aperture. The whole 
image is therefore very messy and much more muddled with subsidiary 
rings and detail. Nevertheless if we take a photograph with the full 
aperture first and know where a particular star suspected of being a 
doublet is located, a subsequent photograph with the apodizing disc 
in place may settle the question of whether or not it is a doublet while 









APPLICATIONS AND RESULTS 
Amplitude 



149 




Fig. 6.15. Principle of apodization. The thin curves are the amplitude 
distributions in the Airy disc patterns of the whole aperture and of an 
aperture of the size of the opaque disc. The thick line is their difference and 
shows that the central disc is much smaller than that for the whole aperture — 
but the first ring is greater in amplitude. 



giving many other spurious effects which we know from the first 
photograph may be ignored. 

6.5. Radio telescopes 

I n the early 1 930s the attention of radio engineers became fixed on the 
origins of some of the curious background noises that were picked up 
on radio receivers and which did not seem to have an origin in known 
terrestrial transmitters. Some of these noises were quickly traced to 
the electrical discharges in thunder storms and the first-ever radio 
telescope was probably a rotatable aerial designed to locate the storm 
centre giving rise to ' atmospherics ' as the cracklings were called. In 
the course of this work it was found that there were variations in 
background noise which had a regular period of about a day. Careful 
studies then showed that the period of variation was about 4 minutes 
less than a solar day of 24 hours — it was in fact tied to the sidereal 
day, that is the day measured in terms of the apparent rotation of a 
particular star through a complete cycle round the Earth rather than 
of the Sun. This careful observation showed that the source of the 
background noise in radio transmissions must be associated with a 
particular region of space. 

Since that time an enormous number of developments have 
occurred and many famous radio telescopes — such as the 250 ft steerable 
telescope at Jodrell Bank — have been built. Fig. 6.16 shows this 



150 



IMAGES 



APPLICATIONS AND RESULTS 



51 



impressive structure which makes a striking object on the sky line as 
one drives north through Cheshire on the M6 motorway. 

The physical principle of the radio telescope is exactly the same as 
that of the optical one. The common form is a large reflecting dish 
which behaves in exactly the same way as the paraboloidal mirror in a 
reflecting telescope. This dish samples the scattering pattern, which is 
usually radiated by sources rather than being scattered by them, and 
the resolving power is calculated in terms of the size of the Airy disc 
of the aperture. 




Fig. 6.16. General view of 250 ft steerable radio telescope at the Nuffield 
Radio-Aslronomy Laboratory, Jodrell Bank, Cheshire. By permission of 
the University of Manchester Nuffield Radio Astronomy Laboratories. 

One important wavelength of radiation coming from interstellar 
matter is at 0-21 m. This is about 3 x 10 5 larger than wavelengths in 
the visible region, and so to match the resolving power of a small 
amateur reflecting telescope for visual astronomy of say 015 m 
diameter we should need a radio telescope dish 45 km in diameter! 

This is not in fact quite as ridiculous as it sounds for the following 
reason. A radio telescope does not produce a picture of a radio 









source directly as does an optical telescope. With the dish fixed at one 
position and pointing in one direction the antenna placed at the focal 
point of the dish records a particular level of signal. Now of course 
the interest may be in the time variations of this signal — as was the 
case of the early observations mentioned above. But if the spatial 
distribution is of interest the telescope must be made to change its 
direction or to move its position on some kind of railway track. What 
is then explored is the radiation pattern of the radio source at the 
Earth's surface and from the appropriate measurements the charac- 
teristics and shape of the source can be deduced. In particular if two 
small telescopes are set up, one fixed and the other moving along a 
straight railway track, it is not only possible to compare the intensity 
at different points in the diffraction pattern but also comparisons in the 
relative phase can be made. This is the advantage of working with 
frequencies of only about 10 9 Hz involving time measurements in 
nanoseconds which are perfectly possible. (Visible light, you will 
remember, has frequencies in the 10" Hz region.) Thus a fairly 
accurate record of the diffraction pattern along this straight line can be 
built up and from this a precise picture of the radio source can be 
produced. There is no reason why such a track should not be 45 km 
long and this relatively simple system would give a resolution along this 
particular line comparable with that of 45 km diameter telescope! 

There are many variations on this theme. Some telescopes are kept 
fixed to the Earth's surface and merely make use of the Earth's rotation 
in space to trace out the radiation patterns. Others use a large number 
of small dishes dotted about over a large area to sample the radiation 
pattern and hence to build up the required pictures. 

6.6. Schlieren techniques 

In our discussion of the Foucault knife-edge method of testing telescope 
mirrors, the point was made that various kinds of deviation from a 
perfect spherical mirror such as dust, scratches, grease, etc., as well as 
actual errors in the shape are revealed. It is not difficult to see that, in 
a similar way, any departures from uniformity in the optical behaviour 
of the medium in between the mirror and the focal point would have 
similar effects. Thus if we set up the system for the knife-edge test 
with an absolutely perfect and clean spherical mirror and adjust the 
knife edge so that the field Just goes dark we then have a sensitive 
system for detecting variations in the refractive index of the air between 
the mirrors and the focal point. Fig. 6.17 shows the use of the system 
to study the initiation of an edge-tone (such as occurs in an organ pipe) 
using a rising column of hot air. 



152 



IMAGES 










Fig. 6.17. (a) and (b) Successive stages in the formation of an ' edge ' tone 
by a column of air rising and incident on an edge. The changes in density 
of the air are revealed by the Schlieren technique. 






APPLICATIONS AND RESULTS 



153 



This kind of system was first introduced for examining the perfection 
of slabs of optical glass. The problem in producing optical glass is 
that the melting point is close to the point at which the material of the 
lining of the ladle dissolves in the glass: if the temperature is just too 
low, the glass is very viscous and it is difficult to remove tiny bubbles; 
if the temperature is just too high, constituents of the ladle lining may 
go into solution and produce regions of different refractive index. 
Since the glass is usually stirred before pouring these regions become 
streaks in the glass and the word Schliere is simply the German for 
' streak '. 

Perhaps the most spectacular application of the technique has been 
in studying the behaviour of model aircraft in wind tunnels where the 
Schlieren technique can be used to reveal the variations in refractive 
index of the air flowing over the aerofoil surface, which of course are 
related to the velocities and densities of the air. Special systems are 
needed; a common modification of the basic Foucault system is 
shown diagramatically in fig. 6.18. Two mirrors are used and again 
it can be seen that the system satisfies the conditions for producing an 
intensity distribution corresponding to the Fraunhofer diffraction 
pattern of whatever is put in the parallel beam. The knife-edge 
eliminates the bright Airy disc which would transform back to the 
uniformly bright field and the diffraction pattern of any fine details 
gets past the edge of the blade and recombines to form images of the 
disturbances in which we are interested. The question is sometimes 
raised that the knife-edge cuts out half the diffraction pattern of even 
the parts of the image in which we are interested. This is true and 
indeed the resultant appearance of the field is different from that which 
would result if we replace the knife-edge by an opaque dot exactly the 



I n\ source 
o s Itgtv 




Wind 'unne: ( 
or ether 

system jnde* I 

observation i 



"Off oxis" 

paraboloidai 

mirror 



Fig 6.18. 
L 




; - axis" 
poraooloidai 



Knife edge 
Diagram of a typical Schlieren system using two large mirrors. 



154 






IMAGES 



size of the Airy disc. But the detail is clearly visible and the defects 
in the image are offset by the enormous increase in difficulty of locating 
the small opaque dot! 

In a very beautiful and useful modification of the Schlieren tech- 
nique, the knife-edge, which can be thought of as a strip of opaque 
material and a strip of transparent material with a straight line border, 
is replaced by a strip of red colour filter and a strip of green colour 
filter with a straight line border. The central Airy disc passes half 
through one and half through the other and hence when the recombina- 
tion takes place the image of the aperture is a mixture of red and green 
light, that is yellow. But any detail in the field which produces 
diffraction effects in the green filter area will be reproduced in green 
and those giving diffraction detail in the red area will be reproduced 
in red. With care it is possible to relate the precise colour to the 
variation in refractive index of the medium and so the picture can be 
interpreted quantitatively. 

6.7. Image-processing 

You will remember that in Section 5.3, in which we discussed Abbe's 
theory of microscopic vision, it became clear that the final image (and 
the principle applies to any image-forming system, not just to micro- 
scopes) is not necessarily a reproduction of the object: rather it is a 
reproduction of a hypothetical object which would give a scattering 
pattern identical with the modified pattern which is actually allowed to 
proceed to the recombination stage. It thus becomes obvious that 
seeing is not believing, as by modifying the scattering pattern in differ- 
ent ways we can make the image almost anything we want. 

Fig. 6.19 shows an example which was prepared using the laser 
diffractometer described in Section 2.6, with the adaptation to permit 




APPLICATIONS AND RESULTS 



155 




Fig. 6.19 (com. opposite). 



(g) (h) 

Fig. 6.19. (a) Object made up of transparent and opaque regions, (b) Centre 
region of diffraction pattern of (a) showing only 3 repeats in the vertical 
direction: in the full pattern there are several more repeats above and below 
those shown, (c) Central unit only of (b). (d) Recombination of (c). 
(e) Central unit of (b) and all others except the adjacent one above and 
below the centre eliminated. (/) Diffraction pattern of (e): note the spacing 
of the stripes relative to those in (a), (g) All units left in vertical direction 
but laterally the pattern is restricted to a very narrow vertical strip, (h) Re- 
combination of (g). From An Atlas of Optical Transforms, by G. Harburn, 
C. A. Taylor and T. R. Welberry, by permission of G. Bell & Sons Ltd. 



156 



IMAGES 



the diffraction pattern to be modified and then recombined; the 
schematic diagram of fig. 2.20 (a) showed the system. The object 
6.19(a) is an outline of a man crossed by stripes. This particular 
object is chosen because striped pictures commonly arise as a result of 
television transmissions, for example, from space-craft. The object 
can be thought of as the product of the man's outline and the set of 
stripes. Thus its diffraction pattern (6.19 (b)) will be the convolution 
of the diffraction pattern of the outline with that of the stripes; the 
latter is a set of points (the so-called ■ orders of diffraction ') arranged 
in a vertical row and so the odd-shaped diffraction pattern of the man's 
outline is repeated many times in a vertical row (fig. 6.19(A) shows 
only the 3 centre repeats). Fig. 6.19 (c) shows the central version on 
its own with the rest masked off; when recombined therefore there is no 
information about the stripes and they disappear (fig. 6.19 (^).) If 
the central pattern is eliminated and just the one on either side allowed 
to contribute (fig. 6.19 (e)), then since these two are twice as far apart, 
the stripes appear to be of half the width! (fig. 6.19 (/)). Finally, if 
all the orders are left in but the width of each is severely restricted 
(fig. 6.19(g)) then the stripes remain but the outline is largely lost 
(fig. 6.19(A)). 

There are all kinds of practical applications of this technique which 
can be far more sophisticated than the one described here. For 
example, if an imaging system has some known defects, it may be 
possible to modify the scattering pattern in such a way as to compensate 
for these defects and so produce a better image. 

A word of caution is necessary though. Image-processing techniques 
may make an image far more acceptable to the eye but they cannot 
actually increase the amount of information present. A crude 
illustration of this might be that if a television picture is produced of a 
man with a pimple on his chin and it so happens that the pimple is 
completely covered by a dark stripe and no trace of it is visible in the 
light stripe on either side, then no amount of image filtering will ever 
reproduce the pimple! This may seem facetious but the warning is 
serious and it is surprising how many good physicists are sometimes 
misled into believing that they can extract more information from 
processed pictures than is there before processing. 

There are, of course, some systems in which processing can actually 
lead to the extraction of more information— but it is very important to 
think clearly about what is actually happening before reaching con- 
clusions. For example, a television picture from a space vehicle may 
be badly affected by noise producing a strong ' snow-storm ' effect over 
the picture. If the picture being transmitted is a still one so that there 






APPLICATIONS AND RESULTS 



157 






is no change in the object over many frames it is quite possible to 
introduce electronic filtering based on comparisons between successive 
frames. Data which appear in two successive frames are preserved 
and any which appear in only one frame are eliminated. If this 
process is repeated over a period very great enhancement of picture 
quality is possible. We are in fact using one of the principles of 
focusing that was elucidated in the first chapter. We know that the 
picture we want is unchanging; we know that the noise that we do not 
want is changing in a random way. This knowledge enables us to 
separate the two. 

6.8. Optical microscopy 

I have used optical microscopy to illustrate numerous aspects of 
imaging theory already in the course of the book and, of course, a 
special section was devoted to the Abbe theory of microscopic vision 
(Section 5.3). There are, however, two special modifications of 
microscopes that have not so far been mentioned and which are splendid 
applications of imaging theory that I think merit inclusion. I have also 
included — largely because I am a great admirer of the quite remarkable 
work done by microscopists in the late ninteenth century — a historical 
reference that provides an interesting example of indirect recombination 
used in optical microscopy. 

The first modification involves the technique known as dark field 
illumination. It is, in fact, very closely related to the Foucault knife- 
edge test and to the Schlieren techniques already described. In a nor- 
mal microscope the object is illuminated by a converging cone of light 
from the illuminating system below the microscope stage. In the dark 
field system, an opaque disc of suitable size is placed in the centre of 
the illuminating system so that the cone of illumination becomes 
hollow. If the stop is of the right size, the field of view on looking 
down the microscope is entirely dark when no object is in place. 
When an object is introduced it can be seen outlined in light against the 
dark background. The system can best be understood by thinking of 
Abbe theory. In effect the stop has removed the central order but 
since, if there is no object, the central order is the only one present in 
the diffraction pattern, the recombination results in a blank field 
(fig. 6.20). Any object in the field however will create a diffraction 
pattern which will get past the opaque disc and recombine to give a 
picture of detail resembling a Schlieren pattern. The technique is 
particularly useful when highly transparent objects are being viewed. 
Differences in thickness and refractive index may be so small that in 



158 



IMAGES 



APPLICATIONS AND RESULTS 



159 




Object 
slide 



Bock focal 

plane of 

objective 




Fig. 6.20. Illustration of the principle of dark-field illumination based on 
Abbe theory. Sets of parallel beams fall in many different directions on to 
the slide and continue to a focus in the back focal plane of the objective from 
which they go forward to be recombined into an image. Introduction of the 
opaque disc to eliminate beams in the axial direction results in the absence 
of a zero order. 

the normal microscope arrangement they can hardly be seen. Varia- 
tions in thickness of transparent material do however give rise to 
diffraction effects and under dark field conditions the details can often 
be seen quite clearly. The technique is not as useful as it might at first 
appear to be because it is extremely difficult to interpret the images 
produced. For example it is impossible to distinguish the thicker 
from the thinner pieces of material. 

The second modification is the much more useful phase-contrast 
system. A full description would be out of place and it will be sufficient 
to say that the principle is similar to that of dark field illumination 
except that instead of eliminating the central order, in phase contrast 
systems the phase of the central order is changed by 90°. In practice 
there are all kinds of problems connected with depth of focus and 
other complications which make various compromises necessary 
but the systems can be made to work and their great advantage is that 
quantitative interpretation is possible; thicker sections of transparent 
material can be made to appear darker than thinner sections. 

* In order to satisfy curiosity I shall try to explain how the two systems 
of dark field and phase contrast work using the phasor diagram (see 
Section 2.4), but I must warn you that this limited explanation is not by 
any means completely satisfactory. 

For the purposes of the explanations we imagine that we have a 
completely transparent object which varies very slightly in thickness and 






hence varies the relative phase of the light passing through it by a small 
amount. In an unmodified microscope it will produce a diffraction 
pattern which will obviously have a large central peak (because so much 
light is passing through the whole object) and relatively weak details 
further from the centre. When recombination occurs the final result 
must be an image with nearly constant amplitude across the field (because 
the object is transparent) but with a slight phase change of which we are 
not aware. Thus we could represent the amplitude at a particular point 
in the field of view by means of the phasor OP in fig. 6.21 (a). The 
phasor OP' would represent the amplitude at a place where the object is 
very slightly thicker and OP" the phase where it is slightly thinner. Since 
the central peak is so strong and can be regarded as a reference beam, 
OP, OP' and OP" can all be resolved into two components as shown in 
fig. 6.21 (b) in which OQ is the central peak and QP, QP' and QP" are 
in each case the contributions of all the rest of the diffraction pattern 
outside the central peak to the point in the object which is of interest. 

We can now represent the effect of dark field by eliminating OQ. You 
can now see that, since OP' is longer than QP, the thicker part will be 
slightly brighter: unfortunately however QP" is the same length as QP' 





(c) 



•■ 





(d) 



Fig. 6.21. Phasor diagrams illustrating dark field and phase contrast micro- 
scopy, (a) OP represents the amplitude at a particular point in the field of 
view: OP' represents the amplitude where the specimen is slightly thicker 
and OP" where it is slightly thinner, (b) Resolution of phasors into com- 
ponents: the effect of dark field illumination can be judged by eliminating 
the central order, OQ. (c) OQ changed in phase by 90° to give positive 
phase contrast, (d) OQ changed in phase by 90° in the opposite direction 
to give negative phase contrast. 



160 



IMAGES 






APPLICATIONS AND RESULTS 



161 



and so the slightly thinner part is also slightly brighter— hence the defects 
of dark field as mentioned above. 

The effect of phase contrast is obtained by turning OQ through 90° 
(fig. 6.21 (c)). Now OP' is shorter than OP so the thick part appears 
darker; OP" is longer than OP and the thinner part appears brighter. 
This is known as positive phase contrast. If the phase shift of 90° had 
been in the other direction the contrast is reversed as can be seen from 
fig. 6.21 (d) which illustrates negative phase contrast. 

Fig. 6.22 gives illustrations of the effect achieved under phase-contrast 
conditions. 

Now I shall turn to my historical interlude. In the 1870s micro- 
scopists used diatoms in a great deal of their experimental work 
because they had extremely regular features (see for example fig. 5.8) 
and were of such a size that many of the details of their structures were 
just about on the limit of optical resolution: they are still often used as 
test objects to assess the perfection of microscope objectives. One 
well known diatom used at that time, and still today, is Pleurosigma 
angulatum. It has an essentially hexagonal structure and was known 
to give orders of diffraction in the back focal plane of the objective 
(Abbe theory) as shown in fig. 6.23 (a). 




(a) 
Fig. 6.22 (com. opposite). 




Fig. 6.22. (a) Normal micrograph ( X 1000) of a cotton hair that has been 
immersed in a solvent which removes the cellulose component and leaves 
the cuticle or ' skin \ (/>) Phase-contrast micrograph ( X 1000) of the same 
specimen as in (a). The slight change in shape arises because between the 
two photographs being taken a little more of the core has been dissolved; 
nevertheless the enhancement of the contrast is very clear. By permission 
of Mr. S. Simmens, Shirley Institute, Manchester. 

A Mr. Stephenson records in the Journal of the Royal Microscopical 
Society for 1878 the following incident. A mathematical student who 
had never seen a diatom, taking the spectra alone, recorded in the back 
focal plane of the microscope objective, (fig. 6.23 (a)) worked out by 
calculation the drawing, reproduced as fig. 6.23 (/>), as the object that 
gave rise to such spectra. Fig. 6.23 (b) is thus his mathematically 
"focused * image of the diatom— an extremely early example of the 
process of indirect recombination! 

Mr. Stephenson goes on to say that the small markings between the 
hexagons had never been seen in P. angulatum by anybody. But on 
his making extra careful investigations and stopping out the central 
pencil " so that its superior illuminating effects might not over-power 
the others " (i.e. he used dark field illumination) " these small markings 



162 



IMAGES 







<«) 



(b) 



Fig. 6.23. (a) Spectra observed in the back focal plane of a microscope 
objective when a diatom is placed on the specimen table, (b) Calculated 
image deduced by a student in the 1870's. From Light, by Lewis Wright 
(1882). 

were found to exist, though they were so faint as to have eluded all 
observations until mathematical calculation from their spectra had 
shown that they must be there." Lewis Wright reporting this in the 
1890s comments " Light was once more, even in the microscope, by 
its physical deportment, a Revealer of what the microscope had, up to 
that date, failed to see." 

6.9. Electron microscopy 

In Section 3.5 we discussed lenses for electrons and indicated some of 
the practical problems met in trying to do with electrons what the 
optical microscope does with photons. The most familiar and well 
established kind of electron microscope is the one known as the 
transmission electron microscope. It can only be used with specimens 
up to about 5 x 10" 7 m thick and the information imaged is the differ- 
ential absorption of electrons by different thicknesses of material and 
by variations in the nature of the specimen from point to point. 
Although there are practical complications (resulting from the need for 
the whole system, including the specimen, to be in a vacuum, for the 
high voltage to be very precisely stabilized, for the electrons to be 
accelerated to sufficiently high energies to penetrate the specimen, and 
from the fact that the lenses are precision-made electromagnetic 
systems), the optical arrangement is identical with that of an optical 



APPLICATIONS AND RESULTS 



163 



microscope. A source of illumination (the electron gun) followed by 
a condenser lens (electromagnetic) irradiates the specimen with elec- 
trons. An objective lens (electromagnetic) then produces a real and 
highly magnified image and a projection lens (also electromagnetic) 
produces another real image with further magnification on a fluorescent 
screen which is also inside the vacuum. For record purposes a 
photographic plate can be inserted through a vacuum trap to take the 
place of the fluorescent screen. 

As is the case with most sophisticated instruments, there is a great 
art in getting the best out of electron microscopes and the mere record 
of the best resolution obtained does not really give an adequate 
impression of the results achieved. To my mind some of the most 
beautiful and revealing photographs ever produced are those by R. G. 
Wyckoff of virus crystals. They are now quite old and have been 
surpassed in terms of fineness of resolution but remain in a class by 
themselves as a combination of scientific significance and artistry. 
Fig. 6.24 is a typical example. 










Fig. 6.24. Electron micrograph of a shadowed replica of a tobacco necrosis 
virus crystal. Magnification approximately 50 000. By permission of 
Professor R. W. G. Wyckoff, University of Arizona. 



164 



IMAGES 



APPLICATIONS AND RESULTS 



165 




Hg. 6.25. Electron micrograph of a thin crystal of magnesium fluoro- 
germanate. The main periodicity of the unit cells in the horizontal direction 
is 0-59 nm and faults in the perfection of the crystal can also be clearly 
seen. The magnification is approximately 3 million. By permission of 
Sumio I.gima, Department of Physics, Arizona State University. 

One very recent example of extremely high resolution electron 
microscopy published in 1976, which achieves a magnification of 
about 3 million times, is shown in fig. 6.25. The material is magnesium 
fluorogermanate and each white square represents a feature of a single 
unit cell of the structure; they are spaced 5-9 x 10 ,0 m apart in the 
horizontal direction. The particular fascination is that here we can 



see very clearly that the crystal is not perfect, and there are stacking 
faults occurring at anything between 3 and 18 cells apart. 

Surfaces of materials are of great interest to scientists but you will 
have realized that the normal transmission microscope is unlikely to be 
able to deal with specimens such as surfaces of fractured metal unless 
the remaining metal supporting the surface can be cut or ground down 
to a thickness of the order of 5xl0 _7 m (about the wavelength of 
visible light by coincidence). Some remarkably elegant ways of 
surmounting this difficulty have been achieved over the years. For 
example, it is possible to make a replica of the surface being studied 
by an exactly similar process to that used by a sculptor in reproducing 
a bronze statue from a clay or plaster mould. A liquid synthetic resin 
mixed with a hardener is painted on to the specimen and, in due course, 
when it has solidified, the replica in very thin plastic is peeled off. Now 
we have a specimen thin enough not to absorb too many electrons. 
Unfortunately, of course, it does not now absorb many at all and so it 
is not able to impress any particular information (in the form of a 
scattering pattern) on the incident electrons if it is placed in the electron 
microscope. 

A second technique known as ' shadowing ' is then brought into 
play. The replica is placed in a vacuum and a thin layer of metal is 
deposited on it by the so-called ' evaporation ' process. If the metal is 
raised to the melting point, atoms of the vapour will escape from the 
surface (just as water molecules evaporate from a water surface well 
below its boiling point) and, because it is in a vacuum, there will be no 
collisions with gas molecules and the atoms will travel in straight lines. 
By a suitable geometrical arrangement (fig. 6.26), the metal atoms can be 
made to fall at an angle on the replica (R) and hence produce ' shadows ' 
in metal. The ups and downs of the surface are thus converted into 
opaque and transparent metal areas and the shadowed replica now has 
the characteristics needed for producing good electron scattering 







Fig. 6.26. Principle of the metal shadowing technique. 



166 



IMAGES 



patterns and then recombined images. Fig. 6.27 shows an electron 
micrograph of a metal surface made in exactly this way. 

But in spite of all the ingenuity in techniques and in spite of the 
apparently favourable wavelength of electrons (50 kV electrons have a 
wavelength of about 5 x 10 -12 m) instrumental errors still prevent the 
electron microscope from achieving its full promise. 




Fig. 6.27. Shadowed replica of the surface of a specimen of carbon steel: 
the protruding ridges are of cementite and the magnification is about 
10 000. By permission of Dr. John Taylor, Department of Metallurgy, 
University College. Cardiff. 



Lens imperfections are one of the principle defects and these arise 
from the fact that the precision with which metal components of the 
electromagnetic lenses need to be made greatly exceeds the optical 
perfection with which glass lenses need to be polished; so far the 
practical problems of achieving this accuracy for the particular shapes 
involved in the electron microscope lenses have not been solved. 

In addition it is very easy for an electron microscope to slip out of 
adjustment. For example the focus might not be quite right or a 
deposit of tungsten evaporated from the filament may fall asymetrically 
on to one of the tiny aperture stops which restrict the electron beam 
and may produce very strange astigmatic effects. In principle what 
happens is that the source of electrons is no longer effectively a point 






APPLICATIONS AND RESULTS 



167 






source but rather acquires some odd shape. The final image is a 
convolution of the true image with the shape of the source and this 
can differ seriously from the true image alone. Further, unless you 
know exactly what the object should look like, you may be unaware 
that defects exist. Figs. 6.28 (a), (c) and (e) show three photographs 
of the same object with three different focusing adjustments and each 
made with a non-astigmatic system. Clearly the pictures are different, 
but without the comparison who is to tell whether either on its own is 
right or wrong? Fortunately, we can use a technique based on 
imaging and diffraction theory to test each picture. 

We have already seen that the image produced is the convolution of 
the true image with a point spread function (see Section 5.5), which is an 
image of the electron source which may well be distorted. Suppose 
now we place the electron micrograph itself as a diffracting object in 
the optical diffractometer. The result will be the product of the 
diffraction pattern of the true image and the diffraction pattern of the 
shape of the image of the source. Now this source image is much 
smaller than the details of the image in which we are interested: it 
follows that the diffraction pattern on the spot is much larger than that 
of the details of the image. Thus the dominant feature of the diffraction 
pattern of the electron micrograph is the diffraction pattern of the spot 
shape and this can easily be interpreted. If the spot is circular and 
tiny the diffraction pattern is large and Airy-disc like. Figs. 6.28 (b), 
(d) and (/) show the diffraction patterns of 6.28 (a), (c) and (e) respec- 
tively. It is not difficult to see that the focus is different and fig. 6.28 (g) 
and (//) show the effect when astigmatism is present. 

This sort of ' image analysis ' is becoming increasingly popular and 
is very helpful in preventing total misinterpretation of the electron 
micrographs. 

The next problem in trying to push forward the limits of what can 
be done with the electron microscope is that it is not easy to distinguish 
in a transmission micrograph between changes in contrast (black, 
white and grey) that come from changes in specimen thickness and 
those that come from changes in chemical nature. A grain of the same 
material on the surface providing extra thickness might for example 
look the same as an impurity inside the crystal of the same size and 
shape. The system known as the microprobe analyser goes some way 
to resolving the difficulty. As the beam of electrons hits any particular 
point of the specimen characteristic X-rays are emitted and their wave- 
length gives information about the material producing them. By first 
producing a normal electron micrograph and then irradiating areas 
of the specimen about which further information is needed with 



168 



IMAGES 



APPLICATIONS AND RESULTS 



169 



(a) 



(c) 



(«) 



(g) 



■'.•' .... ."-■. ■'■•-." 

1 

-•■• . . ■:■■' <Vi-'-.* *, • 





(« 




(</) 



(/) 



(/') 






electrons and analysing the resultant X-rays, great strides in interpreta- 
tion have been made. 

Finally we must talk of the problem of depth of focus introduced in 
Section 5.5. In that section we saw that scanning the object with a 
narrow pencil of electrons gets over the depth problem. But what 
actually happens when the electrons strike the object; are they merely 
scattered? The answer is that they are not and indeed the charges 
that are collected and which form the signal are in fact secondary 
electrons that are emitted from the body of the material. It becomes 
clear then that the specimen is really acting as a self-luminous source, 
but, of course, only the spot being irradiated at any one moment is 
emitting the secondary electrons. 

Careful investigation shows that the angle of the surface affects both 
the penetration of the primary electrons and also the way in which 
secondary electrons can diffuse out, and for this and various other 
reasons we find excellent representations of surfaces produced by the 
scanning electron microscope. The resolution of scanning microscopes 
has progressively been increased. Among other techniques the actual 
size of the primary source of electrons has been reduced by using a fine 
tungsten wire with a tip of radius 5 x 10 -8 m as a field emission source 
(as for the field emission microscope). High energy electrons can be 
used — though of course there comes a point at which the law of 
diminishing returns comes in, because higher energy electrons create 
more damage to the specimen and so the thing we are studying may be 
irretrievably damaged before the image can be examined. Magnifica- 
tions of up to a million in the scanning microscope are, however, quite 
possible and the probability of ' seeing ' individual atoms is just around 
the corner. 

6. 1 0. X-ray, electron and neutron diffraction 

In Chapter 4 we saw that the whole range of techniques of investigation 

that are grouped under the general heading of X-ray diffraction or 

Opposite: 

Fig. 6.28. (a), (c) and (e) Electron micrographs of amorphous carbon film 
with a magnification of about li million and three different adjustments of 
focus, (b), (d) and (/) The corresponding optical transforms of (a), (c) and 
(e). (g) Electron micrograph of amorphous carbon film at a magnification of 
about 5 million but with an astigmatic condition. (/») Optical transform of 
(g) : the oval shape of the central peak reveals the astigmatism. Photographs 
(a), (c) and (e) by Miss D. Chescoe, A.E.I. Harlow; (b), (d), (/) and (h) by 
Dr. D. Somerford, University College, Cardiff; (g) by F. Sheldon, Pye- 
Unicam, Cambridge. 
M 



170 



IMAGES 



X-ray crystallography can be quite properly considered as aspects of 
image formation. With electrons, it is perfectly possible, as we saw in 
the last section, to focus images; but there are some occasions when, 
for a variety of reasons, it can be more useful to interpret electron 
diffraction patterns directly instead of allowing the recombination to 
occur in the electron microscope. In particular this is true when the 
detail being studied is of atomic size and the aberrations of the image- 
recombination system introduce resolution limits that are much coarser 
than the theoretical ones for electron wavelengths. The basic principles 
of interpretation of electron diffraction patterns are broadly similar to 
those for X-ray diffraction, though there are inevitably some differences 
in detail. It is interesting in passing to note that electron diffraction 
was not discovered until about 1 5 years after the discovery of X-ray 
diffraction. 

Neutrons may also be used but here, because they are uncharged, 
there is no way of recombining them directly and interpretation of the 
diffraction pattern is the only avenue open. The special advantage of 
neutrons is that — again because they are uncharged — they are not 
scattered by the electron cloud round an atom but are scattered by 
atomic nuclei. The scattering effects are similar for all atoms and do 
not, as in the case of X-rays, depend on the atomic number. Thus 
hydrogen atoms, which are difficult to locate by X-ray diffraction 
techniques, may be located relatively easily by neutron diffraction. 

In our discussions of X-ray diffraction in Chapter 4 we concentrated 
on illustrating the technique of direct interpretation of the scattering 
pattern without recombination. But, of course, one alternative 
approach is not to bother about interpretation at all, but merely to 
regard the scattering pattern as a kind of ' finger-print ' which can be 
used for identification without interpretation. This is the basis of the 
so-called ' powder ' method that has had enormously successful 
applications over the years. A high proportion of solid materials, 
even when existing as finely divided powder, has a very pronounced 
regularity of structure and consists essentially of collections of tiny 
crystallites each of which is really a complete crystal covering many unit 
cells. Each of these tiny crystallites would give an X-ray diffraction 
pattern consisting of a set of sharp spots on a regular array (such as 
fig. 4.2 (a)). Since a powder consists of many thousands of such 
crystallites in all possible orientations, each crystallite will give the 
same pattern, but patterns will occur in all possible orientations. The 
resulting diffraction pattern will thus consist of sets of concentric rings 
whose diameters and intensities will be characteristic of the material. 
Fig. 6.29 (a) shows such a photograph of a material in which the 



APPLICATIONS AND RESULTS 



171 



crystallites are relatively large and not too numerous; the separate 
spots building up the rings can still be seen. Only short segments of 
the rings are recorded. The material here is coarsely crystallized 
corundum — A1 2 3 — such as is used for grinding lenses. In 6.29 {b) 
the material is the same but the crystallite size is much smaller and the 
spots fuse together into smooth rings. In 6.29 (c) the more usual form 
of photograph on a strip of film is shown ; here the material is the 
mineral rutile (titanium dioxide, Ti0 2 ). 



(a) 
(b) 
(c) 
W) 



I I I 




Fig. 6.29. Four 4 powder ' diffraction photographs, (a) Coarse sample of 
corundum (Al 2 3 ). (b) Very finely divided sample of corundum, (c) and (d) 
Two chemically indistinguishable samples of titanium dioxide. The crystal 
structures are quite different as is revealed by the diffraction patterns, 
(c) is rutile and (d) is anatase. 



It is important to realize that each photograph of this type is not 
characteristic of a particular chemical structure but of the particular 
crystalline or structure variant of the material too. Fig. 6.29 (d) is 
also of titanium dioxide, Ti0 2 and the specimen is chemically identical 
with that for 6.29 (c), but here the crystalline form is that for the 
mineral anatase and the totally different powder diffraction pattern is 
quite obvious. 

One example of purely routine use which immediately jumps to my 
mind relates to the glass industry. In Section 6.6 we mentioned that 
certain kinds of glass, if raised to too high a temperature, attack the 
lining of a furnace or ladle. If a piece of solid material is found in a 
batch of glass it is clearly important that the source should be tracked 
down as soon as possible. An X-ray photograph of the inclusion can 
be taken in a matter of a few minutes and the nature of the material 
will indicate its source; if there are several ladles or furnaces with 
chemically identical linings it is even possible to include in their 
manufacture different chemical markers which can be immediately 
identified from powder photographs. 

X-ray diffraction does not occur only with highly crystalline materials. 
Indeed some of the most challenging patterns to the interpreter come 



172 



IMAGES 



from materials that are not crystalline at all in the simple sense. The 
vast array of new materials classified under the general terms ' plastics ' 
or ' polymers ' are usually not highly regular crystals but the individual 
long chain molecules which characterize them do have regularity of a 
kind and this imposes itself on their X-ray diffraction patterns. Fig. 
6.30 shows two examples from well known materials and it is clear that 
they neither have the sharp spots on a lattice characteristic of single 
crystals nor the rings characteristic of aggregates of tiny crystallites. 
Nevertheless, by the study of the distribution of the blackening of the 
film, a surprisingly large amount of information about the nature and 
structure of the material can be deduced if the research worker is 
familiar with the relationships between objects and their diffraction 
patterns. 

To complete this selection of X-ray diffraction illustrations, I propose 
to give, in somewhat greater detail than in Chapter 4, an example of a 
comparison between optical and X-ray diffraction patterns for a par- 
ticular material; I hope this may prove valuable as a means of consoli- 
dating firmly the ideas about relationships between objects and their 
diffraction patterns that were established in the main text. 

Fig. 4.11 illustrated the final answer to the solution of the struc- 
ture of hexamethylbenzene deduced many years ago by the classical 
methods of X-ray crystallography. Now we will look at some of the 
steps that might have been taken if this structure had been studied by 
optical analogue methods. 

Fig. 6.31 (a) shows an X-ray photograph taken by the precession 
method (Section 4. 1). Fig. 6.31 (b) shows an idealized diagram of one 
molecule in which all twelve carbon-carbon bonds are assumed to be 
the same length. The inner six carbon atoms form the benzene ring 
and the outer six are the nuclei of the six methyl groups. The hydrogen 
atoms are ignored for our present purposes. 

This particular structure has been chosen because it has only one 
molecule in each unit cell and this greatly simplifies the geometrical 
problems and the argument — but I must hasten to tell you that such 
examples are rare. 

The X-ray picture of fig. 6.31 (a) enables us (knowing the dimensions 
of our apparatus and the wavelength of the X-rays) to deduce 
immediately two of the dimensions of the unit cell of the crystal and, if we 
assume the crystal to be a convolution of one molecule with the lattice, 
the photograph should be the product of the diffraction pattern of the 
lattice with the diffraction pattern of one molecule. The diffraction 
pattern of the lattice is often called by crystallographers the ' recipro- 
cal lattice '. 






APPLICATIONS AND RESULTS 



173 







(a) 




(b) 

Fig. 6.30. X-ray diffraction patterns for (a) a polyester fibre (e.g. ' Terylene '), 
(b) a polyamide fibre (e.g. ' Nylon '). 



174 



IMAGES 



Fig. 6.31 (c) is the diffraction pattern of the molecule as shown in 
fig. 6.31 (b) and we need to compare it with fig. 6.31 (a). Can you 
see six groups of spots near the edge of fig. 6.31 (a) that are not 
arranged as a regular hexagon but as an elongated hexagon ? To help 
identification their position is indicated in fig. 6.31 (d). These could 
correspond to the six circular peaks near the edge of 6.31 (c) if the 
molecule, were distorted in some way. Suppose we tilt the molecule 
so that, in projection, it looks like 6.32 (a). Now we have distorted 







f ' -« 






(a) 



(b) 








O - 






i 



". 



T' ' ■' 



(c) (d) 

Fig. 6.31. (a) X-ray precession photograph of hexamethylbenzene. 
(b) Diagram of one molecule of hexamethylbenzene. (c) Diffraction pattern 
of (b). (d) Location of groups of spots in (a) corresponding to the six peaks 
of Us), 



APPLICATIONS AND RESULTS 



175 




"3 

§7, 



•iX 



8 



00 



176 



IMAGES 



the molecule and its diffraction pattern (6.32 (b)) resembles the arrange- 
ment of strong spots in 6.31 (ci) provided that we also rotate it. Fig. 
6.32 (c) shows it rotated into the right position to match 6.31 (a) and 
we have printed the two pictures on top of each other in 6.32 (d). The 
outer six peaks match reasonably well and so we appear to have 
deduced the orientation of the molecule in space. 

This is by no means the whole answer and, as already said, this is a 
particularly easy example; I hope, however, that it gives some idea of 
the kind of deductions that it is possible to make. It should certainly 
help you to see how some of the relationships between objects and 
their diffraction patterns— the Fourier transform relationship— work 
out and become of great practical value. 

Now let us turn to electron diffraction and consider first how the 
patterns are actually observed. Various special arrangements have 
been devised from time to time but by far the simplest technique is 
exactly parallel with that used by Abbe in his studies of the optical 
microscope. The distribution of intensity in the back focal plane of 
the objective of an optical microscope can be studied by rearranging 
the microscope so that the eyepiece focuses on this plane instead of on 
the conjugate image plane of the object. With an electron microscope 
the projection lens which forms the image is re-adjusted so that it 
images the back focal plane of the objective and the electron diffraction 
pattern then replaces the image on the screen or photographic plate. 
One of the study areas in which this technique has proved particularly 
informative is that of thin film technology. We mentioned vacuum 
evaporation as a means of shadowing replicas of surfaces for use in the 
electron microscope in the last section, ff, in the evaporation chamber, 
two or more metals are evaporated simultaneously and allowed to form 
a thin film on a base, or substrate as it is usually called, (it is often a 
polished rock-salt, quartz or other crystal surface) then the resulting 
film is a very special kind of alloy. It is special because it has been 
built up in such a way and the atoms of the various metals present have 
a degree of choice ' in how they arrange themselves in the layers. In 
the more usual processes of crystallization from molten metals the 
build up is much more rapid and, of course, grows outwards from 
nucleation centres in the solid instead of in successive two-dimensional 
layers as in evaporation. The electrical and magnetic properties of 
these films have importance in many branches of modern electronic 
technology, but their structures have been worked out in great detail 
from electron diffraction studies. Fig. 6.33 shows two examples of 
electron diffraction patterns which are both beautiful because of their 
high symmetry and also very full of structural information. It is 



APPLICATIONS AND RESULTS 



177 



important to notice that for an electron diffraction pattern produced 
by readjusting a typical microscope, with an accelerating voltage of say 
50 000 V, the electrons have a de Broglie wavelength of about 
5xl0- 12 m. This is about 1/100 of typical atom-atom spacings in 
solid matter and this kind of wavelength ratio compares with the 
wavelength-to-object size ratio commonly occurring in optical diffrac- 
tion. Our electron diffraction pattern is therefore of relatively small 
angular extent and is very closely parallel in geometry to the optical 
diffraction pattern produced in the diffractometer illustrated in fig. 2.20. 
Although there are many other applications of electron diffraction 
that are important I just want to include one further development. 
A very significant group of researches uses electrons of very small 
energy. The field is usually known by the acronym LEED (Low 
Energy Electron Diffraction). The accelerating voltages used in 
practice are in the region 10-500 V and the corresponding de Broglie 
wavelength is 4x 10~ 10 to 5x 10 u m. This is in the same size range 




(fl) 
Fig. 6.33 (cont. overleaf). 



178 



IMAGES 




(b) 

Fig. 6.33. (a) Electron diffraction pattern of a thin film of an alloy of gold 
and zinc, (b) Electron diffraction pattern of a thin film of an alloy of gold 
and manganese. By permission of Dr. W. S. Michael, Department of 
Physics and Mathematics, Manchester Polytechnic. 

as atom-atom spacings and hence the scattering geometry is likely to 
be much more like that of X-ray diffraction where diffraction through 
angles all the way up to 180° occurs. The experimental arrangement 
for LEED is therefore quite different from that for conventional 
electron diffraction. The schematic arrangement is shown in fig. 6.34. 
The gun irradiates the specimen and the scattered and secondary 
electrons pass back through a system of grids which permit them to be 
accelerated sufficiently to create luminous spots on the phosphor on the 
screen without changing their direction of travel. With such low 
energies it is the surface structure that is revealed in the luminous pattern 
on the screen. Gas and other impurity atoms on the surface presented 
great difficulties in the early days but very high vacuum systems and 
special cleaning techniques have been developed to such a stage that 
very useful surface information may now be obtained. Surface 



APPLICATIONS AND RESULTS 



179 



Fluorescent screen 



Accelerator 
\f* grid 



•- Electrons 






Specimen 



Viewing 
window 



Fig. 6.34. 
system. 



Schematic diagram of a Low Energy Electron Diffraction (LEED) 



behaviour plays an important role in many modern technologies- 
catalysis, corrosion, solid state devices etc., and LEED has already fed 
in a great deal of valuable information for workers in these and other 
fields. 

The wavelengths corresponding to neutrons of thermal energies are 
much the same as those of the X-ray range. For example, at 0°C the 
neutron wavelength would be l-55xl0 _10 m and at 100°C it would 
be l-33xlO- 10 m. The common X-ray wavelength used in crystal- 
lography — the characteristic radiation from a copper target — is l-54x 
10 _10 m. The actual techniques used in neutron diffraction studies 
resemble in principle those used for X-ray diffraction. A collimated 
beam of neutrons emerging from a nuclear reactor falls on a crystal 
and the diffracted beams are detected and measured by means of an 
electronic counter. The interpretation of the scattered beams then 
follows a similar trial-and-error process to that used for X-rays. The 
scale of the apparatus is much larger: the collimator for example has to 
be very massive indeed in order to absorb all neutrons but those going 
in the required direction. The neutron flux from the reactor is not very 
high and relatively large crystals are needed to give easily detected 
beams. The protective shielding required on all parts of the apparatus 
increases the size and mass enormously and this influences to some 
extent the kind of materials and problems that can be studied. 

Neutron diffraction is rarely used on its own. In general one starts 
with an X-ray diffraction study and then, having extracted as much 
information as possible from that a neutron study can be made to give 
very useful additional information which is frequently complementary 
to the X-ray data. 

I said earlier that neutrons are scattered largely by the nucleus and 
not by the electrons. This is not universally true and there are in fact 



180 



IMAGES 



very important interactions with electrons in the case of magnetic 
materials. Indeed, perhaps the most important contribution that 
neutron diffraction has made to the study of solids has been from 
studies of magnetic materials. The detailed study of the degree of 
order in ferromagnetic and antiferromagnetic alloys made possible 
by neutron diffraction is one of our principal sources of information 
on the atomic mechanism of magnetic behaviour. What makes this 
possible is the fact that an atom with its magnetic moment directed 
(say) upwards behaves differently towards neutrons from one with its 
moment directed downwards. Thus a crystal of an alloy of gold and 
manganese might have manganese atoms at the corner of each cubic 
unit cell and a gold atom at the centre of each. X-ray diffraction 
studies would reveal this quite easily (compare the discussion on zinc 
sulphide in Section 4.1). 

A neutron diffraction study however would distinguish between 
manganese atoms having magnetic moments in different directions, and 
a comparison of the two sets of results would reveal a great deal about 
the magnetic distribution of the manganese atoms. 

Among other important contributions from neutron scattering are 
the detection of ' light ' atoms in the presence of much heavier ones. 
X-ray diffraction runs in to difficulties, for example, in detecting 
something like oxygen in the presence of heavy atoms like tungsten or 
gold. The scattering from the heavy atoms is so great that it completely 
masks the effect of the lighter ones. The differences in scattering for 
neutrons are very small and many otherwise difficult problems have 
been resolved by comparison of the two techniques. Work on the 
structure of ice, for example, in which X-rays are hardly able to reveal 
hydrogen positions, work on uranium hydride, potassium fluoride and 
many other compounds of light and heavy atoms has been possible 
with neutron scattering. 



6.11. Radar 

We take radar very much for granted these days, and, apart from 
noticing the spinning or rocking antenna on ships, hovercraft and 
airport control centres, and assuring ourselves that our aeroplane can 
fly quite happily in dense cloud because of its radar, most of us give 
it little thought. It fits very well in to our scheme of image producers 
however and deserves a few pages to itself. 

It has its origins almost as far back as the discovery of radio com- 
munication. It was very soon found that the geometry of a radio 
antenna determined the direction from which it could receive the 



APPLICATIONS AND RESULTS 



181 



strongest signals. An antenna mounted on a rotating turntable 
could therefore give an indication of the direction from which radio 
signals were coming. The second world war, of course, provided 
enormous stimulus to developments of this technique and very soon 
instead of just locating the direction from which a signal came, signals 
were being radiated from a transmitter and the reflections or scattered 
radiation from surrounding objects — particularly aeroplanes — could 
be picked up and their direction determined. Once the idea of trans- 
mitting the signal outwards and detecting the scattered radiation on 
return had been grasped it immediately became possible to determine 
the distance of the scattering object as well as its direction; in effect all 
that is necessary is to time the delay between initial transmission and 
receipt of the returned wave. Radio-location had arrived. The 
American term radar was soon adopted: it is a contraction of RAdio 
Direction And Range \ 

In principle there are several ways in which the essentials of a radar 
system can be arranged. It is almost invariable to use a very short 
burst or pulse of radio waves as the signal in order that the timing — 
needed to measure the range— can be as precise as possible; usually too, 
the pulses follow each other at regular intervals, say 500 per second, 
so that changes in range or direction can be followed. The variations 
are: a pulse can be ' broadcast ' in all directions and then a receiving 
antenna can rotate to find the direction from which the required 
reflection is coming; the transmitted pulse can be beamed in precisely 
the same direction as that in which the receiving antenna is trained and 
the two can be rotated or rocked simultaneously; the system may use 
quite long radio waves which will travel huge distances and make 
detection possible from far away, but inevitably the resolution is very 
poor; very short waves can be used which will give highly detailed 
images of the surroundings but are limited in range by the curvature of 
the Earth just as light beams would be— though of course radar beams 
of this kind are not obstructed by mist, cloud or fog; the display of the 
final result may be in a ' technical ' form which requires an expert to 
interpret it or it may be presented just as though it was a television 
picture which is much easier to understand immediately. 

In spite of all these variations you can see that the basic idea con- 
forms precisely to our image-forming principles: the object is irradiated 
and the required information extracted from the scattered radiation. 
The first point to consider then is the interaction of the incident 
radiation and the object, and this depends chiefly on the wavelength 
selected. Long-range radar might use radiation with a wavelength of 
tens of metres in order to avoid the so-called ' optical limitation ' ; that 



182 



IMAGES 



APPLICATIONS AND RESULTS 



183 



is to be able to ' see ' beyond the horizon. Our usual diffraction 
principles tell us, however, that such interactions will be able to give 
only very crude indications of the presence or absence of an object of 
the size of a small aeroplane, with no detail whatever. As the wave- 
length used becomes shorter, so the resolution increases and systems 
operating with wavelengths of a few millimetres can be made to give 
very detailed images. 

The second question to consider is that of whether direct recombina- 
tion is possible. There are two difficulties. It is possible to refract 
short radio waves — indeed there are school experiments on microwaves 
in which a large lens made of paraffin wax is used to focus a beam. To 
make such a lens sufficiently free from aberrations would be quite a 
challenge but, even if the lens problem could be solved, there are two 
further difficulties. First, the magnitude of the returned signal is 
usually quite small and very considerable amplification is needed 
before it can be satisfactorily detected and it would be a very difficult 
problem to do that simultaneously for all parts of the scattered wave. 
Second, we do not have a photographic film or a photo-sensitive surface 
(such as is used in a television camera) that will detect the whole of a 
two-dimensional pattern simultaneously. The solution adopted is to 
use the scanning principle. The antenna may be some kind of highly 
directional array (like a television aerial) for the longer wavelengths or a 
parabolic reflector or horn at the shorter wavelengths. Often the 
same antenna is used both for transmission and reception with a rapid 
electronic switching device that will prevent the outgoing pulse from 
swamping the receiving circuits. In the crudest kind of system a 
linear cathode-ray oscilloscope scan is triggered at the moment a given 
pulse is transmitted and travels horizontally from left to right. The 
highly amplified reflection is fed to the vertical deflection plates and the 
distance of the object producing it can be deduced from the known 
speed of radio waves and the known rate at which the cathode ray 
tube beam is travelling in the X direction. The whole system is 
synchronized so that the pulses go out at, say, 500 times per second, 
and for each the trace on the tube will be re-produced. The antenna 
can be rotated until a particular peak or ' blip ' is of maximum height 
and then both its direction and distance are known. 

In sophisticated systems, the antenna is made to scan the field of 
view either by rotation or by rocking, or in some other combination of 
movements and the cathode ray tube display is simply arranged so that 
the electron beam makes movements on the screen which relate to the 
movements of the antenna. Fig. 6.35 (a) shows a schematic diagram 
of the cruder type and fig. 6.35 (b) shows a radial scan system which 












Fig. 6.35. (a) Diagram of radar with simple linear scan, (b) Diagram of 
radar with simple radial scan. 



gives a rather more detailed picture of the surroundings. Fig. 6.36 
shows a model radar system using a rotation scan and a radial trace 
but in this case with an ultrasonic beam instead of radio waves. It 
was built more or less to a design published in Wireless World in 1968 
and gives an interesting display of objects in a lecture theatre. In the 
photograph an assistant is showing how separate reflections from his 
body and from his two hands are produced and the relative movements 
of his hands can also be displayed. 

In addition to the obvious military and navigational uses, radar has 
played a considerable part in aerial surveying operations where it has 
two advantages over photography using visible light. The first is that 
mist, fog and clouds are penetrated very easily and do not hold up 
activities, and the second is that the reflection coefficients for different 
crops or land features are quite different from those of visible light 
or of infra-red. It is often possible therefore to identify features on 
large photographs taken from a very considerable height. Fig. 6.37 (a) 
shows a radar image of farmland in Kansas, in which, for example, the 
patches of almost white appearance are in fact fields of sugar beet. 
Fig. 6.37 (b) shows a radar image of part of the San Andreas fault in 
which the rock structures can clearly be seen but the dense vegetation 
is transparent at this particular wavelength and so does not obscure 
the features which geologists could not otherwise see. 



184 



IMAGES 





(«) 



(b) 





(c) 



id) 



Fig. 6.36. (a) Model ultrasonic radar system, (b) System in operation, 
(c) Scan showing location of man and hands as in (b). (d) Scan with 
position of hands only changed. 



APPLICATIONS AND RESULTS 



185 




(a) 




(b) 

Fig. 6.37. (a) Radar image of farmland in Kansas: the almost white patches 
are fields of sugar beet, (b) Radar view of the San Andreas fault in California 
showing formation of rock structures underlying vegetation which does not 
reflect at the particular frequency used. By permission of Westinghouse 
Corporation, Washington, D.C. 
N 



186 



IMAGES 



6. 12. Medical imaging techniques, including the use of X-rays, ultrasonics, 
gamma rays and infrared rays 

Most people have been X-rayed at some time or other, but of course the 
usual radiograph taken for diagnostic purposes is sometimes not regar- 
ded as an image but as a shadow or silhouette. Nevertheless, it 
conforms to our usual principle of scattering and recombination with 
the same geometry as that of the field-emission or field-ion microscope 
(Section 3.2). The principal difference is that in the microscopes the 
object is very close to the source in order to achieve large magnifications, 
whereas for radiographs the object is close to the screen and produces 
an image practically the same size as the object (fig. 6.38 (a)). In order 
to improve the contrast and clarity, a special screen is sometimes 
placed between the object and the film; in effect it consists of a series of 
holes in a lead screen which run parallel to the direction of the X-rays 
from the target of the tube and which will cut out any scattered radia- 
tion which is not going in the ' right ' direction. The principle diffi- 
culty in interpreting X-ray photographs is that there is no depth 
discrimination; the blackening of the film depends purely and simply 
on, and is inversely proportional to, the total absorption between the 
source and the film. It is immaterial whether the object is a very thin 
strong absorber or a very thick weak absorber — the result depends only 
on the total absorption. Similarly the radiograph does not distinguish 
between objects at different levels (fig. 6.38 (b) and (c)). 




(a) 
Fig. 6.38 (cont. opposite). 



APPLICATIONS AND RESULTS 



187 




(b) 




(c) 



Fig. 6.38. (a) Photograph of foam block with various metal objects inserted 
at different levels, (b) Normal radiograph of (a), (c) Side-view radiograph 
showing different levels. By permission of Professor K. T. Evans, Welsh 
National School of Medicine. 



188 



IMAGES 






APPLICATIONS AND RESULTS 



189 



In recent years a number of ingenious systems have been introduced 
which to some extent allow these problems to be surmounted. Two 
will be described here. The first is the so-called tomograph. A 
mechanical transport system is introduced for both X-ray tube and film. 
The patient remains fixed, but during an exposure the X-ray tube travels 
in the opposite direction to the film (fig. 6.39 (a)). If the speeds are 
carefully adjusted a line joining the source to the film can be made to 
rotate about a point at a given level in the patient; then for a plane 
through the patient at that level there will be no relative motion 
between the shadow and the film and hence a clear radiograph is 
produced. Material above and below this plane produces totally 
blurred images which can be ignored (fig. 6.39 {b), (c) and (d)). 

X-roy tube moves this 

^ way and always points 

al the required section 





(C) 



Patient remains stationary 






X-ray film moves this 
way in synchronism 



<«) 







ib) 




(d) 



Fig. 6.39. (a) Schematic diagram of tomograph system, (b), (c) and (d) 
Tomographs picking out different objects by imaging different levels of the 
object of 6.38 (a). By permission of Professor K. T. Evans, Welsh National 
School of Medicine. 



190 



IMAGES 



The second system permits the production of very detailed sectional 
views through quite thick structures — such as the human trunk. The 
technique involves the production of a very large number of standard 
radiographs through the object being studied, each using the X-ray 
beam in the plane of the required section but in a different direction 
(fig. 6.40(a)). We thus have a great deal of information and the 
problem is to disentangle it. The ' focusing ' of the image is done by 
computer and the result is displayed on a television screen. Fig. 
6.40 (b) and (c) show particularly beautiful examples. The process of 
unscrambling the data from the large number of views by computer has 
been illustrated by an analogue suggested by Professor Vainshtein of 
the USSR and a simplified version is presented here. 



X-ray beams poss through ond 
ere -eco'ded in o very lorge number 
Ol directions only two of which ore 
snown 



Section being recorded 
Detector moves across field 




(«) 




(b) 

Fig. 6.40 {cont. opposite). 



APPLICATIONS AND RESULTS 



191 




(<0 
Fig. 6.40. Schematic diagram of the technique used to produce X-ray 
sections, (b) Section through chest showing bone, muscle and fat with the 
heart as the central feature, (c) Section through abdomen showing the liver 
on the left, the pancreas in the middle and the spleen on the right. By 
permission of E.M.I. Central Laboratories. 



Fig. 6.41 {a) shows a pattern of four spots with six directions indica- 
ted. Fig. 6.41 (/)) shows ' radiographs ' using two of these directions. 
Fig. 6.41 (c) shows a ' smear ' pattern derived from (b); the grey lines 
represent loci of possible positions that could be deduced from the 
radiographs. The intersections are locations that would fit both 
radiographs. Fig. 6.41 (d) shows the process applied to six directions 
and the intersections now give unambiguous locations for all four spots. 
Clearly a larger number of directions of projection would greatly 
increase the resolution. The computer, in effect, superposes the data 
from the large number of directional projections in precisely the same 
way. 

At first this seems to be a totally new method of imaging, but in fact 
it does conform to our principles. We included scanning as a direct 
recombination process using point-by-point assembly because usually 
— as for television for example — the process is so rapid that the 
appearance to an observer is the same as if the whole picture had been 
imaged simultaneously as with a lens. In this example we are scanning 
— but it would be more correct to describe this as an indirect recombina- 
tion process. The scanning here is * angle-by-angle '. Each X-ray 






192 



IMAGES 



APPLICATIONS AND RESULTS 



193 



(P) 




(O 





id) 




Fig. 6.41. (a) Simple object of four points with six viewing directions 
indicated, (b) Projections of the object along two of the chosen directions. 
Opposite: (c) Superposition of two patterns which represent possible locations 
of the points of (a) derived from the projections (b). (d) Superposition of 6 
patterns, each derived from a projection in one of the directions indicated in 
(a) : the positions of the original points are clearly revealed. With very much 
greater numbers of projections the precision would obviously increase. 







194 



IMAGES 



APPLICATIONS AND RESULTS 



195 



photograph is taken at a different angle and each can be regarded as 
giving a block of information. The big difference is that we cannot 
present the result without first combining this information by means of 
a computer program. The ' known fact ' that enables us to do the 
recombination or focusing is that every X-ray photograph in all the 
different directions corresponds to a single object which is the same for 
each. 

The dangers associated with exposure of human tissue to X-radiation 
have become increasingly worrying and various techniques have been 
tried out as a substitute. Perhaps the most successful is that of 
ultrasonic scanning, which has proved particularly valuable in examina- 
tions during pregnancy. The technique is simple in principle— though 
realization in practice presents problems, largely because of the 
difficulty in achieving sufficiently high levels of radiation. In early 
systems the patient lay on a table over a tank of water with the portion 
of the body to be scanned in contact with the water surface through a 
hole in the table (fig. 6.42 (a)). An ultrasonic generator directs a beam 
of ultrasonic radiation through the water on to the patient and the 
scattered radiation is picked up by a suitable receiver in the water. 
The incident beam scans the patient and the resulting picture is built up 
point by point. The system is rather like a miniature radar system; 
for any given direction of incidence there will be reflections from all 
the interfaces between different materials, e.g. between water and skin, 
between flesh and bone, between the wall of the uterus and the fluid 
inside, etc. The reflections arise because the velocity of ultrasonic 
waves is different in the different media and so in effect the refractive 
index changes. Thus for each direction of the beam, a series of spots 
on the presentation screen will be produced and, as the angle of inci- 
dence is changed to scan the patient, the picture will be built up. 
Modern systems are simpler; the transmitter/receiver is held in contact 
with the body using a liquid lubricant to ensure acoustic contact. 
Figs. 6.42 (b) and (c) show examples of results. Fig. 6.42 (c) is a 

Opposite: 

Fig. 6.42. (a) Diagram of an early ultrasonic scanning system, (b) Ultra- 
sonic scan of the modern ' gray scale ' type in which variations of absorption 
show up as variations in the gray tone: this scan is of a patient lying on 
her back, scanned longitudinally along the centre line and shows clearly 
the skull of the baby in her womb (marked fs) and its trunk (marked ft), 
(c) ' Echo ' type scan in which the white lines represent boundaries between 
various tissues, liquids, cavities, etc.: the patient is again horizontal and 
scanned at a slight angle to the centre line. The skulls of twins can be seen 
very clearly. By permisison of Picker Corporation. 




Section through potient 

Water-'iiied chamber 
below couch 



(«) 



Tronsmilte' Ond receiver work from 
side lo side ond travel perpendicular 
lo the drawing m order to complete 
scan 



(6) 




(C) 



-ssessws^j. 




* /"S .--' r r*T*\5 









1% 



IMAGES 



particularly striking example in which the reflection from the heads of 
twins inside the womb can clearly be seen. This is a particularly good 
example of our process of scattering and recombination in which the 
recombination is by direct point-by-point assembly. 

For both X-rays and ultrasonics we irradiated the object and collec- 
ted the scattered radiation. We shall now turn to two examples in 
which the objects are ' self-luminous '. 

The first is of growing importance in diagnosing many different 
diseases and corresponds to the use of radioactive tracers in industrial 
applications. The patient either ingests or is injected with a chemical 
which includes a radioactive isotope. The chemical is chosen so that it 
will be concentrated by the body in the organ to be examined (e.g. 
radio-iodine concentrates in the thyroid gland) and of course the half- 
life of the isotope and the dose are selected to ensure minimal danger 
from the radiation. The result however is that the body contains a 
' self-luminous ' organ emitting gamma radiation. In order to recom- 
bine the radiated beams to produce an image, a scanning process is used 
which involves the principle of ' straight-line ' imaging. In principle 
a long, fine hole in a massive tube, say of lead, will allow the passage of 
gamma rays in only one direction. By directing the tube to different 
points of the body in turn a picture of the organ concerned can be 
built up. It is usual nowadays to automate the system so that the 
radiation received by the tube in any direction is recorded by a 
scintillation counter and the whole record is processed by a small 
computer and presented, usually in colour, on a television type screen. 
Finally in this section, it is worth mentioning thermography. The 
human body radiates infrared radiation the whole time, but the amount 
of radiation depends on the precise difference between the surface 
temperature and the surroundings. In various diseases, the tempera- 
ture of the body can be slightly changed and hence an image of the 
body built up from its ' self-luminous ' infrared radiation can be useful 
in diagnosis. 

Fig. 6.43 (a) shows a schematic diagram of a thermographic system. 
It again involves recombination of the radiated beams by the point-by- 
point scanning process. Fig. 6.43 (b) shows a typical example of a 
thermogram. More recent systems use what is, in effect, an infrared 
television camera, and by electronic treatment of the signals, can 
display temperature contours as colour variations. 

6.13. Holography 

The principles of holography were outlined in Section 4.3 and provide 
one of the most fascinating examples of the recombination of scattered 



APPLICATIONS AND RESULTS 



197 



Rototing, toothed - 
wheel to. 'chop' signcl 
in order to produce on 
q.C- signol from the 
detector 



Subject — 




Long focai length 
mirror (sometimes plane) 
which oscilloles bock ond 
forth and olso right to left 
to scon subiect 



(a) 




(b) 
Fig. 6.43. (a) Schematic diagram of thermograph system, (b) Thermo- 
gram of patient with an advanced cancer of the left breast. The dark areas 
indicate the high temperature regions. This photograph is really only 
illustrative of the principle as such an advanced cancer would normally have 
been detected by other means. By permission of Dr. D. K. L. Davies and 
Mr. R. E. Toogood, Velindre Hospital, Cardiff. 



198 



IMAGES 



radiation to produce an image. Our intention here is merely to 
describe a few practical examples of applications of holography. 

The first, which I include because it is of particular interest to me, 
is to the testing of components of musical instruments of the string 
family. Violins, guitars, etc. all involve two major components, the 
strings which determine the frequency of the note being produced and 
its relative amplitude and duration, and the body which amplifies the 
sound and in so doing modifies its quality very considerably. The 
vibrational response of the back and belly plates are thus crucially 
important in determining the tone quality. The problem is, however, 
that it is not too easy to determine the resonant response during con- 
struction while it is still possible to make modifications; nor is it easy 
to know what modifications to shape, thickness, etc. are needed in 
order to make a desired change in vibrational characteristics. One 
solution that has been used is to excite the plate with an electro- 
mechanical driving unit and to scatter sand on it: the sand collects 
along nodal lines to form the well known Chladni figures and these in 
turn enable deductions about the characteristics to be made. The 
amplitude of excitation required is however very high— much higher 
than that used when the instrument is played. Hence the results may 
not be meaningful in relation to practical playing problems. Holo- 
graphy provides an answer. 

We set up the violin plate and produce a hologram of it. The system 
is then set up again with the real violin plate coinciding precisely with 
the image of the violin plate recombined by irradiating the hologram 
with a laser beam. If the two do not coincide precisely interference 
figures are seen and if the real violin plate is driven by a quite weak 
oscillator the interference fringes show up the vibrational pattern. 
Fig. 6.44 shows examples of this technique in use. 

The second example is a particularly elegant technique which is a 
very appropriate one with which to end a book on images, since it uses 
one of the most modern image-forming techniques to produce special 
lenses which themselves are to be used for image forming. With 
highly coherent laser light, conventional lenses scatter so much that 
internal diffraction problems produce insurmountable difficulties and 
holographic lenses may be the solution to this problem. 

The principle is based on that of the zone plate (Section 3.4) and 
the hologram (Section 4.3). The problem is first to provide a zone 
plate with sinusoidally varying zones so that it will only have one focal 
length— a point which was explained in Section 4.3. Then we need 
to improve the transmission— even a sinusoidally varying zone plate 
still works by discarding half the incident light. The solution is to 



APPLICATIONS AND RESULTS 



199 




f; r ;* '"■ '■■■■' ! **wf 






■ 



(a) 





(c) (<0 

Fig. 6.44. Reconstructed holograms showing two modes of vibration for 
guitar and cello front plates. The frequencies of excitation are (a) 148 Hz, 
(b) 236 Hz, (c) 174 Hz and (d) 292 Hz. By permission of Dr. Ian Firth, 
Department of Physics, University of St. Andrews. 



200 



IMAGES 



use a. phase zone plate in which the light waves transmitted through the 
various annuli have their phases changed so that all may contribute 
constructively. How then does this differ from an ordinary lens? 
Is not this what a conventional lens does ? The answer is " Not quite ". 
The conventional lens adds various additional path lengths in order to 
bring all paths through the lens to be exactly equal when an image is in 
focus. The phase zone plate keeps the paths unequal but ensures that 
they always differ by a whole number of wavelengths; optically speaking 
therefore no problem arises. The great advantage is that the phase 
zone plate can be extremely thin even if the aperture is very large, and 
this reduces background scattering, especially when used in coherent 
light. 

But how can such a plate be made ? The practical details are complex 
but the principle is simple. In our discussion on holography (Section 
4.3) the idea of the interaction between a spherical wave diverging from 
a point and a plane reference beam giving rise to a zone-plate pattern 
was used. If that pattern is recorded not on a photographic emulsion 
which gives a black-and-white reproduction but in a photo-sensitive 
polymer which becomes soluble to a greater or lesser extent in a 
suitable solvent depending on the amount of exposure to light, it is 
possible to produce a transparent sheet of polymer with exactly the 
desired variations in thickness. This method of producing holographic 
lenses has only arrived at a usable level of perfection very recently but 
is likely to revolutionize laser optics in the near future. 

There is considerable aesthetic satisfaction in seeing an argument 
come round into a full circle and in being able to use in a very practical 
and useful way a principle which, when it was originally introduced, 
was regarded as little more than a fascinating scientific curiosity. 



Further reading 



The general fields of optical and X-ray diffraction and of image formation 
are covered by a bewildering number of books at many different levels. 
This brief selection is intended to give a few starting points for those wishing 
to delve more deeply into the subjects. They are arranged in chronological 
order. 

Taylor, C. A., and Lipson, H. (1964). Optical Transforms. London: G. Bell 
& Sons. 

Pament, G. B., and Thompson, B. J. (1969). Physical Optics Notebook. 
California: Society of Photo-optical Instrumentation Engineers. 

Lipson, G., and Lipson, H. (1969). Optical Physics. Cambridge: The 
University Press. 

Lipson, H. (1970). Crystals and X-rays. London: Wykeham Publications. 

Woolfson, M. M. (1970). X-ray Crystallography. Cambridge: The University 
Press. 

Smith, F. G., and Thomson, J. H. (1971). Optics. London: Wiley. 

Williams, C. S., and Beckland, O. A. (1972). Optics, A Short Course for 
Engineers and Scientists. New York : Wiley. 

Lipson, H. (Editor) (1972). Optical Transforms. London: Academic Press. 

Harburn, G., Taylor, C. A., and Welberry, T. R. (1975). Atlas of Optical 
Transforms. London: G. Bell & Sons. 

Images and Information. Open University Course Text, 1977. 



Index 



Abbe, theory of microscopic vision 

120, 157, 176 
Aberration, chromatic 130 

spherical 123, 130 
Airy disc 25, 111, 113, 115, 138 

144, 153, 168 
Alloys, study of thin fibres by elec- 
tron diffraction 177 
Aluminium oxide, X-ray powder 

pattern 177 
Amplitude and intensity in diffrac- 
tion pattern 25, 27, 50 
Analogue, optical, of emission micro- 
scope 61 
of X-ray diffraction 87 
of heavy atom method 98 
Aperture 108, 134 
and depth of focus 135 
and resolution 108, 115 
Apodization 148 
Astigmatism 123, 166 
Astronomical telescope 111, 140 
Astronomy, radio 111, 149 
Atom, heavy, method in X-ray 

diffraction 96 
Atoms, imaging of 82 



Bandwidth and coherence 21 

of various radiations 23 
Bragg Law 83 
Brain, partnership with eye 2 
Broglie, L. de 24 



Camera 131 

obscura 1 33 

pinhole 55,58,81 

precession 86 
Carbon steel, electron micrograph of 
166 



Coding, as a stage in imaging 3 
Coherence 13 

and bandwidth 21 

effects on diffraction patterns 46 

of laser source 21 

length 16 

measurement of 51 

negative 52 

partial 16 

in practice 1 8 

spatial 1 3 

temporal 14 

of X-rays 82, 85 
Coherent light, images in 54 
Compton scattering 35 
Convolution 56. 112 

and multiplication 1 13 
Copper sulphate, X-ray diffraction 

pattern of 84 
Corundum, X-ray powder pattern of 

171 
Crystallography, X-ray 82, 171 



Dark field illumination 158 
Decoding as a stage in imaging 3 
Demonstration of basic imaging 
ideas 1, 24 

of coherence of radio waves 19 

with water waves 14 
Detection of radiation 9 

Diatom, as microscope object 118 
160 

Diffraction. Fraunhofer 36, 111 
138 
Fresnel 36 
optical, observations of 29, 36, 

40, 47, 49 
nomenclature 35 
X-ray 81, 84, 87, 94, 170 



INDEX 



203 



Diffraction gratings 121 

crossed 1 1 3 

with sinusoidal variation 106 
Diffractometer, optical 46, 154 
Double-slit interference experiment 
35,41 



Electromagnetic spectrum 10 

Electron density 99 
microscope 75, 162 
microscope, scanning 127 
micrograph compared with optical 
micrograph 1 1 8 

Electrons 1 2 

diffraction of 80, 170, 177 
helical patterns in lens 76 
lenses for 76 

Eye 1, 129 



F-number 134 

and depth of focus 135 
False detail in images 115, 117 
Far-field approximation 38 
Fibre, X-ray photograph 172 
Field-emission microscope 61 
Field-ion microscope 61 
Film, grain in 135 
Focus, depth of 124, 135 
Focusing 5 

by computer 97 
Formants, image 122 
Foucault knife edge test 143 
Fourier integration 30 

summation 30 

transformation 30, 37, 43, 97, 
113, 123, 176 
Fraunhofer diffraction 36,111,138 
Frequency of various radiations 1 1 
Fringes, Lloyd's mirror 50 

Young's slits 35, 51 
Fresnel diffraction 36, 63 

half-period zones 74 

zone plate 73, 200 
Function, point-spread 124 

transfer 124 



Galilean telescope 140 
Gamma-rays, imaging with 12, 195 



Glass manufacture, use of X-ray 
diffraction 171 



Hale telescope 142 
Halo, diffraction 40 
Heavy atom method in X-ray diffrac- 
tion 96 
Heisenberg's principle 23 
Hexamethylbenzene 101, 172 
Hologram 2, 4, 102, 198 
Holographic lenses 199 



Illumination, coherent 54 

incoherent 54 
Image, as interference pattern 68 
limits on precision of 7 
processing 1 54 

size of with pinhole camera 56 
two stage with X-rays and light 

101 
X-ray with pinhole camera 57 
Image-forming process 5 
Imaging, by scanning 77 
straight-line 55 
by zone-plate 72 
with coherent light 3, 54 
with y-rays 12, 195 
with incoherent light 3, 54 
with infrared 196 
with radar 185 

with self luminous objects 3. 54 
ultrasonic 27, 192 
Incoherent radiation 7 

imaging 3, 54 
Indirect recombination of scattered 

radiation 82 
Information, present in scattered 
light 2 
form of, in scattered light 7 
transmitted by modulation of 
carrier 21 
Intensity and amplitude in diffrac- 
tion patterns 25, 27, 50 
Interference, Newton's rings 35 
nomenclature 35 
Young's slits 35, 51 
Interferometer, Michelson Stellar 
146 



204 



INDEX 



Jodrell Bank telescop" 150 



Keplerian telescope 140 
Knife edge test 143, 151 



Laser as coherent source 20 

Laue photographs 83 

Lens, basic function of 3 
electromagnetic, for electrons 76 

166 
holographic 200 
imperfections of 123,166 
as phase adjuster 69,130 
relationship to pinhole 65 

Lenses for light 65, 130 

Limit of resolution 8,108 

Limitations of imaging systems 7 
108 

Lloyd's mirror 50 

Low Energy Electron Diffraction 
178 



Masks 25, 88 

Magnesium fluorogermanate, elec- 
tron micrograph of 164 
Mica plates for phase changing 1 00 
Michelson stellar interferometer 1 46 
Micrographs, comparison of optical 

and electron 118 
Microscope, electron 75,162 

field emission 61 

field ion 61 

optical 157 

resolution of 115 

two- wavelength 100 
Microscopic vision. Abbe theory of 

120 
Microwaves, imaging with 182 
Modulation 22 



Neutron diffraction 81, 170, 180 

imaging 12 
Newton's rings 35 
Newtonian telescope 141 
Nipkow disc 78 



Opal 40 

Optical analogues of field emission 
microscope 61 
heavy atom method 98 
methods of solving X-ray diffrac- 
tion problems 89 
X-ray diffraction 87 
Optical diffractometer 46 
Optical and electron micrographs 

compared 1 1 8 
Optical microscopy 115, 157 
Optical transforms 25, 38, 48 

Partial coherence 14 
measurement of 51 

Phase adjuster, lens as 69 
cannot be recorded 25, 48 
changing with mica 100 
contrast illumination 158 
problem in X-ray diffraction 100 
relationships in scattered light 7 

Phasor diagram 34 

Photon momentum 24 

Phthalocyanine 99 

Pinhole camera 55 
camera for X-rays 58, 81 
relationship to lens 65 

Planck's constant 24 

Point source for diffraction 41 

Point spread function 124 

Polymers, X-ray diffraction from 
172 

Powder diffraction with X-rays 171 

Projection of three-dimensional 
studies 88 

Projector experiments with 2, 24 

Radar 180 

Radiation, detection of 9 

scattering of 3 
Radio-astronomy 111,149 
Radio-waves, coherence of 19 

detection of 9 
Radiography 186 
Reciprocal lattice 174 
Reciprocal relationships in diffrac- 
tion 8,43, 113 
Recombination of scattered waves 3 

in practice 109 



INDEX 



205 



Replica techniques 165 
Resolution and aperture 108, 115 



San Andreas fault, radar image of 

185 
Scanning, imaging by 77 
Scattering, Compton 35 

nature of 24 

nomenclature 35 

with non-visible radiation 47 
Schlieren techniques 144, 151 
Schmidt telescope 145 
Self-luminous object, imaging with 

3, 54, 58 
Shadowing techniques 165 
Sharpness of pinhole image 66 
Slits, double 35, 41 
Source, slit and point composed 41 
Sugar beet, radar image 1 85 



Telescope, aperture of 111 

astronomical 140 

Galilean 140 

Hale 142 

Keplerian 140 

Newtonian 141 

optical 140 

radio 149 

Schmidt 145 

terrestrial 140 
Television processing of pictures 
151 

scanning in 78 
Temporal coherence 51 
Thermography 196 
Titanium dioxide 171 
Tobacco necrosis virus, electron 

micrography of 163 
Tomography 1 87 
Transfer function 124 
Translation property of Fourier 
transform 43 



Trial and error structure determina- 
tion 96 
Tungsten, studies of metallic 95 



Ultrasonic imaging 12 
imaging in medicine 192 
radar 184 
waves, coherence of 19 

Uncertainty principle 21, 23 

Unit cell 93 



Violin, holographic interferometry 
of 198 



Water waves 13 

Wavelength in electromagnetic region 
10 

and resolution 108 

of various radiations 1 1 



X-rays, coherence of 82 
crystallography 82 
diffraction patterns 81, 84, 87, 

94, 170 
imaging 9, 12, 81 
powder diffraction patterns 171 
sections 190 
tomograms 187 



Young's double slit experiment 35, 
41 



Zinc sulphide, X-ray diffraction 

pattern 84, 87 
Zone plate 72 

phase 200 
Zones, Fresnel half period 74 



THE WYKEHAM SCIENCE SERIES 



• 



i 

2 

3 

4 

5 

6 

7 

8 

9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 



Elementary Science of Metals J. W. Martin and R. A. Hull 

t Neutron Physics G. E. Bacon and G. R. Noakes 

{Essentials of Meteorology D. H. McIntosh, A. S. Thom and V. T. Saunders 

Nuclear Fusion H. R. Hulme and A. McB. Collieu 

Water Waves N. F. Barber and G. Ghey 

Gravity and the Earth A. H. Cook and V. T. Saunders 

Relativity and High Energy Physics W. G. V. Rosser and R. K. McCulloch 

The Method of Science R. Harre and D. G. F. Eastwood 

t Introduction to Polymer Science L. R. G. Treloar and W. F. Archenhold 

{The Stars; their structure and evolution R. J. Tayler and A. S. Everest 

Superconductivity A. W. B. Taylor and G. R. Noakes 

Neutrinos G. M. Lewis and G. A. Whbatley 

Crystals and X-rays H. S. Lipson and R. M. Lee 

■[Biological Effects of Radiation J. E. Coggle and G. R. Noakes 

Units and Standards for Electromagnetism P. Vigoureux and R. A. R. Tricker 

The Inert Gases: Model Systems for Science B. L. Smith and J. P. Webb 

Thin Films K. D. Leaver, B. N. Chapman and H. T. Richards 

Elementary Experiments with Lasers G. Wright and G. Foxcroft 

[Production, Pollution, Protection W. B. Yapp and M. I. Smith 

Solid State Electronic Devices D. V. Morgan, M. J. Howes and J. Sutcliffe 



Strong Materials 
[Elementary Quantum Mechanics 

The Origin of the Chemical Elements 

The Physical Properties of Glass 

Amphibians 

The Senses of Animals 
f Temperature Regulation 
{Chemical Engineering in Practice 
{An Introduction to Electrochemical Science 



Vertebrate Hard Tissues 
■[The Astronomical Telescope 
Computers in Biology 
Electron Microscopy and Analysis 
Introduction to Modern Microscopy 
Real Solids and Radiation 
The Aerospace Environment 
The Liquid Phase 
[From Single Cells to Plants 
The Control of Technology 
Cosmic Rays 



J. W. Martin and R. A. Hull 

Sir Nevill Mott and M. Berry 

R. J. Tayler and A. S. Everest 

D. G. Holloway and D. A. Tawney 

J. F. D. Frazer and O. H. Frazer 

E. T. Burtt and A. Pringle 

S. A. Richards and P. S. Fielden 

G. Nonhebel and M. Berry 

J. O'M. Bockris, N. Bonciocat, 

F. Gutmann and M. Berry 

L. B. Halstead and R. Hill 

B. V. Barlow and A. S. Everest 

J. A. Nelder and R. D. Kime 

J. Goodhew and L. E. Cartwright 

H. N. Southworth and R. A. Hull 

A. E. Hughes, D. Pooley and B. Woolnough 

T. Beer and M. D. Kucherawy 

D. H. Trevena and R. J. Cooke 



P 



E. Thomas, M. R. Davey and J. I. Williams 

D. Elliott and R. Elliott 

J. G. Wilson and G. E. Perry 

Global Geology M. A. Khan and B. Matthews 

[Running, Walking and Jumping: The science of locomotion A. I. Dagg and A. James 

{Geology of the Moon J. E. Guest, R. Greeley and E. Hay 



[Geology of i 

{The Mass Spectrometer 

{The Structure of Planets 

{images 

{The Covalent Bond 

{Science ivith Pocket Calculators 



J. R. Majer and M. P. Berry 

G. H. A. Cole and W. G. Watton 

C. A. Taylor and G. E. Foxcroft 

H. S. Pickering 

D. Green and J. Lewis 



THE WYKEHAM ENGINEERING AND 
TECHNOLOGY SERIES 



Frequency Conversion J. Thomson, W. E. Turk and M J. Beesley 

Electrical Measuring Instruments E Handscombe 

Industrial Radiology Techniques R. Halmshaw 

Understanding and Measuring Vibrations *• _"• VV allace 

Introduction to Tribology J. Halling and W. E. W. Smith 



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About this book 

Diffraction and imaging processes, no 
matter what radiation is used, have a 
common underlying principle. The 
unifying principle is that of the Fourier 
transform, the theoretical basis of most 
modern optical studies. The approach is 
largely visual, using many original 
photographs, and demonstrates the 
potential and limitations of a wide range of 
diffraction, imaging and image-processing 
techniques without plunging into the 
mathematical theory. 



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About the Wykeham Series 

The Wykeham Science Series is intended 
to provide up-to-date accounts of various 
scientific subjects at a level intelligible to the 
senior secondary student whilst at the same 
time providing valuable material for first- 
year undergraduate courses. Each book 
seeks to reinforce the link between school 
and higher education, and this aim is 
achieved by collaboration between a main 
author, who is a distinguished research 
worker, and a school teacher who has 
brought the experience of many years of 
successful senior school teaching. 



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